Some lectures in basic analysis

Stephen Semmes Rice University

Preface

These informal notes are largely concerned with metric spaces, sequences and series, and continuous mappings. In particular, compactness, completeness, connectedness, and related notions will play central roles.

There are many textbooks in analysis, in which the reader can find much more information. These notes are intended to complement such textbooks, some of which can be found in the bibliography.

The abstract notion of a metric space is one of the main topics discussed in these notes, and is defined in Section 1.1. Basic examples include the real line, the complex plane, and n-dimensional Euclidean spaces, with their standard Euclidean metrics.

Some important properties of the real numbers that will be needed here are discussed in Sections 1.2 and 1.3. Some readers may be familiar with properties like these already, and otherwise more information can be found in many texts, including ones in the bibliography.

Many readers probably have at least some familiarity with sequences and series of real numbers. Here we shall consider convergence of sequences in arbitrary metric spaces, as well as some topics related to infinite series of real and complex numbers. As usual, we shall focus on some of the more theoretical aspects, rather than some of the typical examples that many readers may have seen previously.

One of the motivations for the study of continuity and convergence is the theory behind calculus. Some readers may have some familiarity with this already, and otherwise some of the results discussed here can be used to make precise well-known arguments related to differentiation and integration. Some related matters will be discussed in Chapter 5.

Some additional topics related to those discussed here will be mentioned in Section 4.15. Some related aspects of history may be found in [8, 13, 14, 15, 16, 29, 30, 32, 48, 49, 50, 52, 53, 54, 55, 56, 57, 59, 86, 108], for instance. Some related songs may be found in [90, 91, 92]. Some remarks concerning the clarity of explanations in mathematics may be found in [58].

A question came up about the axiom of choice, and we shall not deal with this much here. The axiom of choice is implicitly used in some proofs, particularly with a sequence of choices. Sometimes the axiom of choice is implicitly used, and a slightly different argument would not require it. A very nice reference for some topics like these is [71]. Some more information may be found in [63, 96], for instance.

Countability and uncountability of sets are discussed at the beginning of Chapter 2, without getting into cardinalities of sets too much. This is discussed further in [71] too. The reader may also be interested in Gödel’s paper [42] on the continuum hypothesis. This is mentioned in [71] as well.

I would like to dedicate these notes to the memory of my undergraduate advisor, Dr. Richard M. Summerville.

Contents

1 Metric spaces 1 1.1 The definition, and some examples . . . . . . . . . . . . . . . . . 1 1.2 Supremum and infimum . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Additional properties of R . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Open balls and open sets . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Limit points and closed sets . . . . . . . . . . . . . . . . . . . . . 5 1.6 Unions, intersections, and closures . . . . . . . . . . . . . . . . . 7 1.7 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.8 Bounded sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.9 Compactness and closed sets . . . . . . . . . . . . . . . . . . . . 12 1.10 Relatively open sets . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.11 Sequences of closed intervals . . . . . . . . . . . . . . . . . . . . . 15 1.12 Compactness of closed intervals . . . . . . . . . . . . . . . . . . . 17 1.13 Cells in R^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.14 Compactness of cells . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.15 Totally bounded sets . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Metric spaces, continued 22 2.1 Countable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 Uncountable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Separable metric spaces . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5 Lindelöf’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6 Countable open coverings . . . . . . . . . . . . . . . . . . . . . . 29 2.7 Getting compactness . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.8 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.9 The Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Sequences and series 34 3.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Convergent sequences . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Sums and products . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Subsequences and sequential compactness . . . . . . . . . . . . . 39 3.5 Cauchy sequences and completeness . . . . . . . . . . . . . . . . 40 3.6 Some particular sequences . . . . . . . . . . . . . . . . . . . . . . 42 3.7 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.8 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.9 Cauchy’s condensation test . . . . . . . . . . . . . . . . . . . . . 48 3.10 Another criterion for convergence . . . . . . . . . . . . . . . . . . 49 3.11 Extended real numbers . . . . . . . . . . . . . . . . . . . . . . . . 50 3.12 Limits superior and inferior . . . . . . . . . . . . . . . . . . . . . 53 3.13 Applications for infinite series . . . . . . . . . . . . . . . . . . . . 55 3.14 Rearrangements of infinite series . . . . . . . . . . . . . . . . . . 57 3.15 Cauchy products of infinite series . . . . . . . . . . . . . . . . . . 58

4 Continuous mappings 60 4.1 Continuity at a point . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Compositions and inverse images . . . . . . . . . . . . . . . . . . 61 4.3 Images of compact sets . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5 Images of connected sets . . . . . . . . . . . . . . . . . . . . . . . 66 4.6 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.7 Continuity of power series . . . . . . . . . . . . . . . . . . . . . . 69 4.8 The supremum metric . . . . . . . . . . . . . . . . . . . . . . . . 70 4.9 Completeness of C_{b}(M, N) . . . . . . . . . . . . . . . . . . . . . . 71 4.10 The limit of a function . . . . . . . . . . . . . . . . . . . . . . . . 72 4.11 One-sided limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.12 Prescribing jump discontinuities . . . . . . . . . . . . . . . . . . . 75 4.13 Path connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.14 Integral metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.15 Some additional topics . . . . . . . . . . . . . . . . . . . . . . . . 78

5 Some derivatives and integrals 80 5.1 Functions on subsets of R . . . . . . . . . . . . . . . . . . . . . . 80 5.2 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3 One-sided derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 More on one-sided derivatives . . . . . . . . . . . . . . . . . . . . 84 5.5 The Darboux property . . . . . . . . . . . . . . . . . . . . . . . . 85 5.6 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.7 Lipschitz mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.8 Suprema and infima of functions . . . . . . . . . . . . . . . . . . 89 5.9 More on suprema and infima of functions . . . . . . . . . . . . . 90 5.10 Darboux sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.11 Darboux–Stieltjes sums . . . . . . . . . . . . . . . . . . . . . . . 91 5.12 More on Darboux sums . . . . . . . . . . . . . . . . . . . . . . . 92 5.13 Upper and lower integrals . . . . . . . . . . . . . . . . . . . . . . 93 5.14 Common refinements . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.15 Riemann–Stieltjes integrals . . . . . . . . . . . . . . . . . . . . . 96 5.16 Continuity and integrability . . . . . . . . . . . . . . . . . . . . . 97 5.17 More on Riemann–Stieltjes integrals . . . . . . . . . . . . . . . . 98 5.18 Multiplying functions by constants . . . . . . . . . . . . . . . . . 98 5.19 Adding two functions . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.20 Another monotonically increasing function . . . . . . . . . . . . . 102 5.21 Some inequalities with integrals . . . . . . . . . . . . . . . . . . . 104 5.22 Subintervals of [a, b] . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.23 Derivatives of integrals . . . . . . . . . . . . . . . . . . . . . . . . 107 5.24 Integrals of derivatives . . . . . . . . . . . . . . . . . . . . . . . . 108 5.25 More on Riemann–Stieltjes integrability . . . . . . . . . . . . . . 110 5.26 Differentiability of α . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.27 Changing variables . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.28 Some sums of step functions . . . . . . . . . . . . . . . . . . . . . 113 5.29 Limits of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Bibliography 116

Index 123

Chapter 1

Metric spaces

1.1 The definition, and some examples

Definition 1.1.1 A metric space is a set M together with a nonnegative realvalued function d(x, y) defined for x, y ∈ M that satisfies the following three conditions. First, for each x, y ∈ M, we have that (1.1.2) d(x, y) = 0 if and only if x = y. Second, d(x, y) should be symmetric in x and y, so that (1.1.3) d(x, y) = d(y, x) for every x, y ∈ M. The third condition is that (1.1.4) d(x, z) ≤ d(x, y) + d(y, z) for every x, y, z ∈ M, which is called the triangle inequality. Under these conditions, d(·, ·) is said to define a metric on M.

As a basic example, the real line R is a metric space with respect to the standard Euclidean metric. If x is a real number, then the absolute value |x| of x is defined as usual by

(1.1.5)  |x| = x when x ≥ 0
= −x when x ≤ 0.

It is well known and not difficult to see that |x + y| ≤ |x| + |y| (1.1.6) for every x, y ∈ R. More precisely, we have equality in (1.1.6) when x and y have the same sign, and otherwise there is some cancellation in the left side. The standard Euclidean metric on R is defined by d(x, y) = |x − y|. (1.1.7)

One can check directly that this satisfies the requirements of a metric on R. In particular, (1.1.4) follows from (1.1.6) in this situation.

Let n be a positive integer, and let R^{n} be the usual space of n-tuples of real numbers. Thus an element of R^{n} can be given as

x = (x_{1}, …, x_{n}), (1.1.8)

where x_{1}, …, x_{n} ∈ R. The standard Euclidean metric on R^{n} is defined by

d(x, y) = ( ∑_{j=1}^{n}(x_{j} − y_{j})^{2})^{1/2}, (1.1.9)

using the nonnegative square root on the right side. It is easy to see that this satisfies the first two requirements (1.1.2), (1.1.3) of a metric. It is well known that (1.1.9) satisfies the triangle inequality (1.1.4), but this is more complicated when n ≥ 2, and we shall not give a proof here. One can check that

d′(x, y) = ∑_{j=1}^{n} |x_{j} − y_{j}| (1.1.10)

and

d′′(x, y) = max_{1≤j≤n} |x_{j} − y_{j}| (1.1.11)

also define metrics on R^{n}. In particular, the triangle inequality for (1.1.10) and (1.1.11) can be obtained from (1.1.6). If n = 1, then (1.1.9), (1.1.10), and (1.1.11) are all the same as (1.1.7).

Let M be any set. If x, y ∈ M, then put

(1.1.12)  d(x, y) = 0 when x = y
= 1 when x ≠ y.

One can check that this defines a metric on M. This is called the discrete metric on M.

Now let (M, d(x, y)) be an arbitrary metric space, and let E be a subset of M. It is easy to see that the restriction of d(x, y) to x, y ∈ E defines a metric on E. Thus E may be considered as a metric space as well, using this metric.

1.2 Supremum and infimum

Definition 1.2.1 Let A be a subset of the real line. A real number b is said to be an upper bound for A if for every a ∈ A we have that a ≤ b. A real number c is said to be the least upper bound or supremum of A in R if it satisfies the follwing two conditions. First,

(1.2.2) c is an upper bound for A.

Second, (1.2.3) if b ∈ R is an upper bound for A, then c ≤ b.

It is easy to see that such a c is unique when it exists. More precisely, if c′, c′′ ∈ R both satisfy these two conditions, then one can check that c′ ≤ c′′ and c′′ ≤ c′. This implies that c′ = c′′, as desired.

An important feature of the real numbers is the least upper bound property. This says that if A is a nonempty subset of R with an upper bound in R, then A has a least upper bound in R. As before, we shall also refer to this as the supremum of A in R, which may be denoted sup A.

The least upper bound property is an important difference between the real line and the set Q of rational numbers. If A is a nonempty subset of Q with an upper bound in Q, then the supremum of A in R may not be an element of Q.

There are well-known ways in which R can be obtained from Q, but we shall not pursue this here.

Definition 1.2.4 Let A be a subset of R again. A real number y is said to be a lower bound for A if for every x ∈ A we have that y ≤ x. A real number z is said to be the greatest lower bound or infimum of A in R if it satisfies the following two conditions. First,

(1.2.5) z is a lower bound for A.

Second, (1.2.6) if y ∈ R is a lower bound for A, then y ≤ z.

One can check that such a z is unique when it exists, as before. In this case, z may be denoted inf A.

If A is not the empty set ∅ and A has a lower bound in R, then A has a greatest lower bound in R. To see this, put

(1.2.7) B = {y ∈ R : y is a lower bound for A}.

Thus B ≠ ∅, by hypothesis, and every element of A is an upper bound for B. The least upper bound property for R implies that B has a least upper bound in R. One can verify that sup B also satisfies the requirements of the greatest lower bound for A, as desired.

Alternatively, let −A be the set of real numbers of the form −a, with a ∈ A. Of course, −A ≠ ∅, because A ≠ ∅. If y ∈ R is a lower bound for A, then −y is an upper bound for −A. Hence −A has a supremum in R. One can check that − sup(−A) satisfies the requirements of the infimum of A.

1.3 Additional properties of R

The archimedean property of the real numbers is the following:

(1.3.1) if x and y are positive real numbers, then there is a positive integer n such that y < n x.

Here is another important property of the real line:

(1.3.2) if a, b ∈ R and a < b, then there is an r ∈ Q such that a < r < b.

It is not too difficult to see that each of these two properties implies that other. There is a well-known argument by which the archimedean property can be obtained using the least upper bound property of R. Alternatively, (1.3.2) can be obtained from standard constructions of R from Q.

Let x be a positive real number, and let n be a positive integer. It is well known that there is a unique positive real number y such that

y^{n} = x. (1.3.3)

The uniqueness of y can be verified directly, but the existence of y is more complicated. Let A be the set of nonnegative real numbers t such that

t^{n} < x. (1.3.4)

Note that 0 ∈ A, so that A ≠ ∅. If x ≤ 1, then it is easy to see that 1 is an upper bound for A. Otherwise, if x > 1, then one can check that x is an upper bound for A. In both cases, A has an upper bound in R, which implies that A has a supremum in R. It is well known that y = sup A satisfies (1.3.3), but we shall not prove this here.

If m is another positive integer, then one can define x^{m/n} to be the positive nth root of x^{m} > 0. Using this, one can define x^{r} when r ∈ Q and r > 0. More precisely, one can verify that this does not depend on the particular representation for r as a ratio of positive integers. Note that x^{r} ≥ 1 when x ≥ 1, and x^{r} ≤ 1 when x ≤ 1.

If t is a positive real number, then one can define x^{t} as a positive real number too. More precisely, if x ≥ 1, then one can take x^{t} to be the supremum of the set of positive real numbers of the form x^{r}, where r ∈ Q and 0 < r ≤ t. Otherwise, if 0 < x ≤ 1, then one can reduce to the previous case, by considering 1/x.

1.4 Open balls and open sets

Let (M, d(x, y)) be a metric space.

Definition 1.4.1 If x is an element of M and r is a positive real number, then the open ball in M centered at x with radius r is defined to be the set of y ∈ M such that d(x, y) < r, i.e.,

B(x, r) = {y ∈ M : d(x, y) < r}. (1.4.2)

A subset U of M is said to be an open set in M with respect to d(·, ·) if for every x ∈ U there is a positive real number r such that

B(x, r) ⊆ U. (1.4.3)

1.5. LIMIT POINTS AND CLOSED SETS 5

Note that M is automatically an open subset of itself. The empty set is an open set in M as well. Proposition 1.4.4 If z ∈ M and t is a positive real number, then B(z, t) is an open set in M.

To see this, let an element x of B(z, t) be given. Thus d(x, z) < t, so that r = t − d(x, z) > 0. It suffices to show that B(x, r) ⊆ B(z, t). (1.4.5) To do this, let y be an arbitrary point in B(x, r), so that d(x, y) < r. Using the triangle inequality, we get that d(y, z) ≤ d(y, x) + d(x, z) < r + d(x, z) = t. (1.4.6) This means that d(y, z) < t, and hence y ∈ B(z, t). This shows that (1.4.5) holds, as desired. Proposition 1.4.7 If z ∈ M and t is a nonnegative real number, then V (z, t) = {w ∈ M : d(z, w) > t} (1.4.8) is an open set in M.

To prove this, let x ∈ V (z, t) be given, so that r = d(x, z) − t > 0. Under these conditions, one can use the triangle inequality to show that B(x, r) ⊆ V (z, t). (1.4.9) The details are left as an exercise.

If a and b are real numbers with a < b, then (a, b) = {x ∈ R : a < x < b} (1.4.10) is the open interval in R from a to b. It is easy to see that this is an open set in R, with respect to the standard Euclidean metric. This is the same as the open ball in R centered at the midpoint (a + b)/2 of the interval with radius (b − a)/2.

1.5 Limit points and closed sets

Let (M, d(x, y)) be a metric space, and let E be a subset of M. Definition 1.5.1 A point p ∈ M is said to be a limit point of E in M if for every positive real number r there is a point q ∈ E such that d(p, q) < r and p ≠ q.

Note that a limit point of E in M may or may not be an element of E.

Proposition 1.5.2 If p ∈ M is a limit point of E, then for each r > 0 there are infinitely many elements of E in B(p, r).

Let r > 0 be given, and suppose for the sake of a contradiction that there are only finitely many elements of E in B(p, r). Let q_{1}, …, q_{n} be a list of the elements of E in B(p, r) that are not equal to p. Put

t = min_{1≤j≤n} d(p, q_{j}), (1.5.3)

which is a positive real number. By construction, there are no elements of E in B(p, t), except perhaps p, contradicting the hypothesis that p be a limit point of E in M.

Definition 1.5.4 A subset E of M is said to be a closed set with respect to d(·, ·) if for every p ∈ M such that p is a limit point of E, we have that p ∈ E.

If E ⊆ M has only finitely many elements, then the previous proposition implies that E has no limit points in M, and hence that E is a closed set in M. Observe that M is automatically a closed set in itself.

If a, b ∈ R and a ≤ b, then the closed interval in R from a to b is defined by

[a, b] = {x ∈ R : a ≤ x ≤ b}. (1.5.5)

It is easy to see that this is a closed set in R, with respect to the standard Euclidean metric.

If A and B are sets, then let A \ B be the set of points in A and not in B, i.e., A \ B = {x ∈ A : x ∉ B}. (1.5.6)

If M is a set and E is a subset of M, then M \ E is called the complement of E in M. In this case, we have that

M \ (M \ E) = E. (1.5.7)

Let (M, d(x, y)) be a metric space again.

Proposition 1.5.8 A subset E of M is a closed set if and only if M \ E is an open set in M.

Suppose first that E is a closed set, and let us check that M \ E is an open set. Let p ∈ M \ E be given. Note that p is not a limit point of E, because p ∉ E and E is a closed set. This implies that there is an r > 0 such that B(p, r) does not contain any elements of E, except perhaps p. It follows that B(p, r) does not contain any elements of E, because p ∉ E. Equivalently, this means that B(p, r) ⊆ M \ E, (1.5.9)

as desired.

Conversely, suppose that M \ E is an open set in M, and let us show that E is a closed set. If p ∈ M \ E, then there is an r > 0 such that (1.5.9) holds, because M \ E is an open set. This is the same as saying that

B(p, r) ∩ E = ∅. (1.5.10)

In particular, this means that p is not a limit point of E in M. It follows that every limit point of E in M is contained in E, as desired.

Using the proposition, we get that a subset U of M is an open set if and only if M \ U is a closed set. More precisely, this follows from the proposition with E = M \ U.

1.6 Unions, intersections, and closures

Let (M, d(x, y)) be a metric space.

Proposition 1.6.1 If U_{1}, …, U_{n} are finitely many open subsets of M, then their intersection ⋂_{j=1}^{n} U_{j} is an open set in M as well.

Let x be an arbitrary element of ⋂_{j=1}^{n} U_{j}, so that x ∈ U_{j} for each j = 1, …, n. Thus, for each j = 1, …, n, there is a positive real number r_{j} such that B(x, r_{j}) ⊆ U_{j}, (1.6.2)

because U_{j} is an open set in M. Put

r = min(r_{1}, …, r_{n}), (1.6.3)

and note that r > 0. Clearly

B(x, r) ⊆ B(x, r_{j}) ⊆ U_{j} (1.6.4)

for each j = 1, …, n, so that

B(x, r) ⊆ ⋂_{j=1}^{n} U_{j}, (1.6.5)

as desired.

Corollary 1.6.6 If E_{1}, …, E_{n} are finitely many closed subsets of M, then their union ⋃_{j=1}^{n} E_{j} is a closed set too.

It suffices to show that the complement M \ ( ⋃_{j=1}^{n} E_{j}) of the union is an open set in M, by Proposition 1.5.8. It is well known and not difficult to check that

M \ ( ⋃_{j=1}^{n} E_{j}) = ⋂_{j=1}^{n} (M \ E_{j}). (1.6.7)

We also have that M \ E_{j} is an open set in M for each j = 1, …, n, because E_{j} is a closed set by hypothesis. Proposition 1.6.1 implies that the right side of (1.6.7) is an open set in M, as desired.

Proposition 1.6.8 Let A be a nonempty set, and suppose that U_{α} is an open subset of M for each α ∈ A. Under these conditions, ⋃_{α∈A} U_{α} is an open set in M.

Let x ∈ ⋃_{α∈A} U_{α} be given, so that there is an index α_{0} ∈ A such that x ∈ U_{α_{0}}. Because U_{α_{0}} is an open set in M, by hypothesis, there is a positive real number r such that B(x, r) ⊆ U_{α_{0}}. (1.6.9)

This implies that

B(x, r) ⊆ ⋃_{α∈A} U_{α}, (1.6.10)

as desired.

Corollary 1.6.11 If A is a nonempty set, and E_{α} ⊆ M is a closed set for each α ∈ A, then ⋂_{α∈A} E_{α} is a closed set in M.

One can verify that

M \ ( ⋂_{α∈A} E_{α}) = ⋃_{α∈A}(M \ E_{α}), (1.6.12)

as before. Note that M \ E_{α} is an open set for every α ∈ A, because E_{α} is a closed set, by hypothesis. Thus the right side of (1.6.12) is an open set too, by Proposition 1.6.8. This implies that ⋂_{α∈A} E_{α} is a closed set, by Proposition 1.5.8.

Alternatively, suppose that p ∈ M is a limit point of ⋂_{α∈A} E_{α}. If β is any element of A, then one can check that p is a limit point of E_{β}, because ⋂_{α∈A} E_{α} ⊆ E_{β}. This implies that p ∈ E_{β}, because E_{β} is a closed set in M. It follows that p ∈ ⋂_{α∈A} E_{α}, because the previous statement holds for every β ∈ A. This shows that ⋂_{α∈A} E_{α} contains all of its limit points in M, as desired.

Definition 1.6.13 Let E be a subset of M. The closure of E in M is the set \overline{E} consisting of all p ∈ M such that p ∈ E, p is a limit point of E, or both.

If E ⊆ M is a closed set, then \overline{E} = E, because E contains its limit points in M. Conversely, if \overline{E} = E, then E contains all of its limit points in M, and hence E is a closed set.

Proposition 1.6.14 If E is any subset of M, then \overline{E} is a closed set in M.

It is enough to show that M \ \overline{E} is an open set in M, as in Proposition 1.5.8. Let p ∈ M \\overline{E} be given, so that p ∉ E and p is not a limit point of E. The latter implies that there is a positive real number r such that B(p, r) does not contain any elements of E, other than p. This means that B(p, r) does not contain any elements of E, because p ∉ E. Equivalently, B(p, r) ⊆ M \ E. Using this, one can verify that B(p, r) ⊆ M \ \overline{E}, (1.6.15)

as desired. Alternatively, one can show that every limit point of \overline{E} in M is a limit point of E, and hence an element of \overline{E}.

1.7 Compactness

Let (M, d(x, y)) be a metric space, and let K be a subset of M.

Definition 1.7.1 A family {U_{α}}_{α∈A} of open subsets of M is said to be an open covering of K in M if

K ⊆ ⋃_{α∈A} U_{α}. (1.7.2)

If every open covering of K in M can be reduced to a finite subcovering of K, then K is said to be compact in M. More precisely, K is compact in M if for every open covering {U_{α}}_{α∈A} of K in M, there are finitely many indices α_{1}, …, α_{n} ∈ A such that

K ⊆ ⋃_{j=1}^{n} U_{α_{j}}. (1.7.3)

It is easy to see that finite subsets of M are compact.

Let us consider a couple of examples in the real line, equipped with the standard Euclidean metric. Consider

K = {0} ∪ {1/j : j ∈ Z_{+}}, (1.7.4)

where Z_{+} denotes the set of positive integers. We would like to show that this set is compact in R. To do this, let {U_{α}}_{α∈A} be an arbitrary open covering of K in R. In particular, there is an α_{0} ∈ A such that 0 ∈ U_{α_{0}}, because 0 ∈ K. This implies that there is a positive real number r such that

(−r, r) ⊆ U_{α_{0}}, (1.7.5)

because U_{α_{0}} is an open set in R. Let n be a positive integer such that 1/r < n, as in the archimedean property for R. Equivalently, 1/n < r. If n = 1, then it follows that K ⊆ U_{α_{0}}. Otherwise, if n > 1, then for each j = 1, …, n − 1, let α_{j} be an element of A such that 1/j ∈ U_{α_{j}}. Under these conditions, we get that

K ⊆ ⋃_{j=0}^{n−1} U_{α_{j}}, (1.7.6)

as desired.

Next, let us show that the open unit interval (0, 1) is not compact in R. Thus we would like to find an open covering of (0, 1) in R for which there is no finite subcovering. One way to do this is to use the family of open intervals of the form (1/j, 1), where j is a positive integer greater than or equal to two. Each of these open intervals is an open set in R with respect to the standard Euclidean metric, and it is easy to see that

⋃_{j=2}^{∞}(1/j, 1) = (0, 1). (1.7.7)

However, one can check that (0, 1) is not contained in the union of finitely many of these intervals. More precisely,

(1/j, 1) ⊆ (1/(j + 1), 1) (1.7.8)

for each j ≥ 2. This implies that the union of finitely many intervals of the form (1/j, 1) is equal to a single such interval.

Let (M, d(x, y)) be any metric space again.

Definition 1.7.9 A subset E of M is said to have the limit point property if for every subset L of E such that L has infinitely many elements, there is a point p ∈ E that is a limit point of L in M.

If E ⊆ M has only finitely many elements, then E automatically has the limit point property.

Proposition 1.7.10 If K ⊆ M is compact, then K has the limit point property.

Let L be an infinite subset of K, and suppose for the sake of a contradiction that L has no limit point in K. Let p ∈ K be given, so that p is not a limit point of L. This means that there is a positive real number r(p) such that B(p, r(p)) does not contain any elements of L, except perhaps p itself. The family of open balls B(p, r(p)) of this type, with p ∈ K, is an open covering of K, because p ∈ B(p, r(p)) for each p ∈ K. If K is compact, then there are finitely many elements p_{1}, …, p_{n} of K such that

K ⊆ ⋃_{j=1}^{n} B(p_{j}, r(p_{j})). (1.7.11)

In particular, it follows that

L ⊆ ⋃_{j=1}^{n} B(p_{j}, r(p_{j})), (1.7.12)

because L ⊆ K, by hypothesis. This implies that L has at most n elements, because p_{j} is the only element of B(p_{j}, r(p_{j})) that can be in L. This contradicts the hypothesis that L have infinitely many elements, as desired.

1.8 Bounded sets

Let (M, d(x, y)) be a metric space.

Definition 1.8.1 A subset E of M is said to be bounded in M with respect to d(·, ·) if there is a point p ∈ M and a positive real number r such that

E ⊆ B(p, r). (1.8.2)

The empty set is considered to be a bounded set in M, even when M = ∅.

Proposition 1.8.3 If K ⊆ M is compact and p ∈ M, then there is a positive real number r such that K ⊆ B(p, r). (1.8.4)

Let p ∈ M be given, and observe that

⋃_{j=1}^{∞} B(p, j) = M. (1.8.5)

It follows that the family of open balls B(p, j), j ∈ Z_{+}, is an open covering of K in M. If K is compact, then there are finitely many positive integers j_{1}, …, j_{n} such that

K ⊆ ⋃_{l=1}^{n} B(p, j_{l}). (1.8.6)

This implies that (1.8.4) holds with

r = max(j_{1}, …, j_{n}), (1.8.7)

as desired.

Proposition 1.8.8 If E ⊆ M has the limit point property and p ∈ M, then there is a positive real number r such that E ⊆ B(p, r).

Suppose for the sake of a contradiction that for each positive integer j,

E ⊈ B(p, j). (1.8.9)

Thus, for each positive integer j, there is an element of E whose distance to p is at least j. Let us now choose, for each positive integer j, such a point x_{j} ∈ E with d(p, x_{j}) ≥ j. (1.8.10)

Let L = {x_{j} : j ∈ Z_{+}} (1.8.11)

be the set of points in E that have been chosen in this way.

Suppose for the sake of a contradiction that L has only finitely many elements. This implies that there is an x ∈ E such that x = x_{j} for infinitely many j. In this case, we would have that

d(p, x) ≥ j (1.8.12)

for infinitely many positive integers j, by (1.8.10). However, the archimedean property for R implies that there is a positive integer n such that d(p, x) ≤ n, so that (1.8.12) holds for only finitely many j ∈ Z_{+}.

Thus L has infinitely many elements. If E has the limit point property, then there is a point q ∈ E such that q is a limit point of L in M. In particular, this implies that B(q, 1) contains infinitely many elements of L, as in Proposition 1.5.2. It follows that d(q, x_{j}) < 1 (1.8.13)

for infinitely many j ∈ Z_{+}, by the definition (1.8.11) of L. Using this and the triangle inequality, we get that

d(p, x_{j}) ≤ d(p, q) + d(q, x_{j}) < d(p, q) + 1 (1.8.14)

for infinitely many j ∈ Z_{+}. Combining this with (1.8.10), we obtain that

j < d(p, q) + 1 (1.8.15)

for infinitely many j ∈ Z_{+}. This is a contradiction, because (1.8.15) holds for only finitely many positive integers j, as before.

1.9 Compactness and closed sets

Let (M, d(x, y)) be a metric space.

Proposition 1.9.1 If K ⊆ M is compact, then K is a closed set in M.

Let p ∈ M \ K be given, and let us show that p is not a limit point of K. If j is a positive integer, then V (p, 1/j) consists of the elements of M whose distance to p is larger than 1/j, as in (1.4.8). Remember that this is an open set in M, as in Proposition 1.4.7. It is easy to see that

⋃_{j=1}^{∞} V (p, 1/j) = M \ {p}. (1.9.2)

This implies that the family of sets V (p, 1/j), j ∈ Z_{+}, is an open covering of K in M, because p ∈ M \ K, by hypothesis.

If K is compact, then there are finitely many positive integers j_{1}, …, j_{n} such that

K ⊆ ⋃_{l=1}^{n} V (p, 1/j_{l}). (1.9.3)

This means that K ⊆ V (p, r), (1.9.4)

where r is the positive real number defined by

1/r = max(j_{1}, …, j_{n}). (1.9.5)

In particular, there are no elements of K in B(p, r), so that p is not a limit point of K, as desired.

Proposition 1.9.6 Suppose that K ⊆ M is compact, and that E ⊆ M is a closed set. If E ⊆ K, then E is compact in M.

To see this, let {U_{α}}_{α∈A} be an arbitrary open covering of E in M. Note that M \ E is an open set, because E is a closed set, as in Proposition 1.5.8. We also have that

K ⊆ M = E ∪ (M \ E) ⊆ ( ⋃_{α∈A} U_{α}) ∪ (M \ E). (1.9.7)

This implies that the collection of U_{α}, α ∈ A, together with M \ E, forms an open covering of K in M.

If K is compact, then there are finitely many indices α_{1}, …, α_{n} ∈ A such that

K ⊆ ( ⋃_{j=1}^{n} U_{α_{j}}) ∪ (M \ E). (1.9.8)

In particular, this means that

E ⊆ ( ⋃_{j=1}^{n} U_{α_{j}}) ∪ (M \ E), (1.9.9)

because E ⊆ K, by hypothesis. Of course, this implies that

E ⊆ ⋃_{j=1}^{n} U_{α_{j}}, (1.9.10)

as desired.

1.10 Relatively open sets

Let (M, d(x, y)) be a metric space, and let Y be a subset of M. Remember that the restriction of d(x, y) to x, y ∈ Y defines a metric on Y, so that Y may be considered as a metric space as well.

Definition 1.10.1 A subset E of Y is said to be relatively open in Y if E is an open set in Y with respect to the restriction of d(x, y) to x, y ∈ Y.

Suppose for instance that M = R^{2}, equipped with the standard Euclidean metric, and Y is a line in R^{2}. One can have open line segments in Y that are relatively open in Y, and not open as subsets of M.

Let (M, d(x, y)) be any metric space again, and let Y be a subset of M. If x ∈ Y and r is a positive real number, then let B_{Y} (x, r) be the corresponding open ball in Y, with respect to the restriction of d(·, ·) to Y. Thus

B_{Y} (x, r) = {y ∈ Y : d(x, y) < r} = B(x, r) ∩ Y, (1.10.2)

where we continue to use B(x, r) for the open ball in M centered at x with radius r. By definition, a subset E of Y is relatively open in Y if and only if for every x ∈ E there is an r > 0 such that

B_{Y} (x, r) ⊆ E. (1.10.3)

Proposition 1.10.4 A subset E of Y is relatively open in Y if and only if there is an open subset U of M such that

E = U ∩ Y. (1.10.5)

To prove the “if” part, suppose that U ⊆ M is an open set, and let us check that E = U ∩ Y is relatively open in Y. Let x ∈ E be given, so that x ∈ U in particular. Because U is an open set in M, there is an r > 0 such that B(x, r) ⊆ U. This implies that

B_{Y} (x, r) = B(x, r) ∩ Y ⊆ U ∩ Y = E, (1.10.6)

as desired.

Conversely, suppose that E is a relatively open subset of Y. Thus, for each x ∈ E, there is a positive real number r(x) such that

B_{Y} (x, r(x)) ⊆ E. (1.10.7)

Put

U = ⋃_{x∈E} B(x, r(x)). (1.10.8)

This may be interpreted as being the empty set when E = ∅. Observe that U is an open set in M, because open balls are open sets, and a union of open sets is an open set too, as in Propositions 1.4.4 and 1.6.8.

We would like to verify that (1.10.5) holds in this situation. It is easy to see that E ⊆ U, because x ∈ B(x, r(x)) for every x ∈ E. This implies that

E ⊆ U ∩ Y, (1.10.9)

because E ⊆ Y, by hypothesis. Using the definition (1.10.8) of U, we get that

U ∩ Y = ( ⋃_{x∈E} B(x, r(x))) ∩ Y = ⋃_{x∈E}(B(x, r(x)) ∩ Y). (1.10.10)

It follows that

U ∩ Y = ⋃_{x∈E} B_{Y} (x, r(x)) ⊆ E, (1.10.11)

as desired, using (1.10.2) in the first step, and (1.10.7) in the second step.

Proposition 1.10.12 Let K be a subset of Y. Under these conditions, K is compact as a subset of M if and only K is compact as a subset of Y, with respect to the restriction of d(·, ·) to Y.

Suppose first that K is compact as a subset of Y, and let us check that K is compact as a subset of M. Let {U_{α}}_{α∈A} be an arbitrary open covering of K in M. If α ∈ A, then U_{α} is an open subset of M, and hence U_{α} ∩ Y is relatively open in Y, by the previous proposition. We also have that

K ⊆ ( ⋃_{α∈A} U_{α}) ∩ Y = ⋃_{α∈A}(U_{α} ∩ Y), (1.10.13)

because K is contained in each of ⋃_{α∈A} U_{α} and Y, by hypothesis. This shows that {U_{α} ∩ Y }_{α∈A} is an open covering of K in Y. If K is compact as a subset of Y, then there are finitely many indices α_{1}, …, α_{n} ∈ A such that

K ⊆ ⋃_{j=1}^{n}(U_{α_{j}} ∩ Y). (1.10.14)

This implies that K ⊆ ⋃_{j=1}^{n} U_{α_{j}}, as desired.

Conversely, suppose that K is compact as a subset of M, and let us show that K is compact as a subset of Y. To do this, let {E_{α}}_{α∈A} be an arbitrary open covering of K in Y. This means that E_{α} is a relatively open subset of Y for each α ∈ A, and that

K ⊆ ⋃_{α∈A} E_{α}. (1.10.15)

If α ∈ A, then there is an open subset U_{α} of M such that

E_{α} = U_{α} ∩ Y, (1.10.16)

by Proposition 1.10.4. This leads to an open covering {U_{α}}_{α∈A} of K in M. If K is compact in M, then there are finitely many indices α_{1}, …, α_{n} ∈ A such that K ⊆ ⋃_{j=1}^{n} U_{α_{j}}. It follows that

K ⊆ ( ⋃_{j=1}^{n} U_{α_{j}}) ∩ Y = ⋃_{j=1}^{n}(U_{α_{j}} ∩ Y) = ⋃_{j=1}^{n} E_{α_{j}}, (1.10.17)

as desired, because K ⊆ Y.

1.11 Sequences of closed intervals

Proposition 1.11.1 Let I_{1}, I_{2}, I_{3}, … be an infinite sequence of closed intervals in the real line. If I_{j+1} ⊆ I_{j} (1.11.2)

for every positive integer j, then

⋂_{j=1}^{∞} I_{j} ≠ ∅. (1.11.3)

If j ∈ Z_{+}, then there are real numbers a_{j}, b_{j} such that

a_{j} ≤ b_{j} (1.11.4)

and I_{j} = [a_{j}, b_{j}], by the definition of a closed interval in Section 1.5. We also have that a_{j} ≤ a_{j+1} and b_{j+1} ≤ b_{j} (1.11.5)

16 CHAPTER 1. METRIC SPACES for every j ≥ 1, because of (1.11.2). Put A = {a_{j} : j ∈ Z_{+}}. (1.11.6) Note that A ≠ ∅, because a_{1} ∈ A.

Let us check that a_{j} ≤ b_{l} (1.11.7) for all positive integers j and l. If j ≤ l, then a_{j} ≤ a_{l} ≤ b_{l}. (1.11.8) Similarly, if j ≥ l, then a_{j} ≤ b_{j} ≤ b_{l}. (1.11.9) Thus, for each positive integer l, b_{l} is an upper bound of A. This implies that the supremum of A exists in R, and satisfies sup A ≤ b_{l} (1.11.10) for every l ≥ 1. Of course, a_{j} ≤ sup A for every j ≥ 1, by construction. It follows that sup A ∈ I_{j} (1.11.11) for every j ≥ 1, as desired.

Put B = {b_{l} : l ∈ Z_{+}}, (1.11.12) which is another nonempty set of real numbers. Using (1.11.7), we get that a_{j} is a lower bound for B for each j ≥ 1. This means that the infimum of B exists in R, and satisfies a_{j} ≤ inf B (1.11.13) for every j ≥ 1. We also have that inf B ≤ b_{l} for every l ≥ 1, so that inf B ∈ I_{j} (1.11.14) for every j ≥ 1. Observe that sup A ≤ inf B, (1.11.15) by (1.11.10) or (1.11.13). One can check that

⋂_{j=1}^{∞} I_{j} = [sup A, inf B]. (1.11.16)

However, this is not needed for Proposition 1.11.1.

1.12 Compactness of closed intervals

Theorem 1.12.1 Let a and b be real numbers, with a ≤ b. Under these conditions, the closed interval I = [a, b] is a compact subset of the real line, with respect to the standard Euclidean metric.

Let {U_{α}}_{α∈A} be an arbitrary open covering of I in R, and suppose for the sake of a contradiction that I cannot be covered by finitely many U_{α}’s. Put

L = [a, (a + b)/2] and R = [(a + b)/2, b], (1.12.2)

which are the closed intervals corresponding to the left and right halves of I. In particular, L ∪ R = I, (1.12.3)

and (1.12.4) length(L) = length(R) = length(I)/2,

where length(I) = b−a is the usual length of an interval. It follows from (1.12.3) that at least one of L and R cannot be covered by finitely many U_{α}’s, because of the analogous property for I. Let us now choose I_{1} to be either L or R, in such a way that I_{1} cannot be covered by finitely many U_{α}’s.

Continuing in this way, we can find an infinite sequence I_{1}, I_{2}, I_{3}, … of closed intervals in the real line with the following properties. First,

(1.12.5) I_{1} ⊆ I and I_{j+1} ⊆ I_{j} for every j ≥ 1.

Second,

(1.12.6) length(I_{1}) = length(I)/2 and length(I_{j+1}) = length(I_{j})/2 for every j ≥ 1.

Third,

(1.12.7) for each j ∈ Z_{+}, I_{j} cannot be covered by finitely many U_{α}’s.

More precisely, suppose that I_{j} = [a_{j}, b_{j}] has been chosen in this way for some positive integer j. Put

L_{j} = [a_{j}, (a_{j} + b_{j})/2] and R_{j} = [(a_{j} + b_{j})/2, b_{j}], (1.12.8)

so that I_{j} = L_{j} ∪ R_{j} (1.12.9)

and (1.12.10) length(L_{j}) = length(R_{j}) = length(I_{j})/2.

Using (1.12.7) and (1.12.9), we get that at least one of L_{j} and R_{j} cannot be covered by finitely many U_{α}’s. We take I_{j+1} to be either L_{j} or R_{j}, in such a way that I_{j+1} cannot be covered by finitely many U_{α}’s.

Note that ⋂_{j=1}^{∞} I_{j} ≠ ∅, by (1.12.5) and Proposition 1.11.1. Let x be an element of ⋂_{j=1}^{∞} I_{j}, which is also an element of I, because I_{1} ⊆ I. This implies that there is an α_{0} ∈ A such that x ∈ U_{α_{0}}, because I is covered by the U_{α}’s. It follows that (x − r, x + r) ⊆ U_{α_{0}} (1.12.11)

for some positive real number r, because U_{α_{0}} is an open set in R. If j ∈ Z_{+} is sufficiently large, then one can check that

(1.12.12) length(I_{j}) = 2^{−j} length(I) < r.

In this case, we have that

I_{j} ⊆ (x − r, x + r) ⊆ U_{α_{0}}, (1.12.13)

because x ∈ I_{j}. This contradicts (1.12.7), as desired.

1.13 Cells in R^{n}

Let n be a positive integer. If A_{1}, …, A_{n} are n sets, then their Cartesian product may be denoted A_{1} × ⋯ × A_{n} (1.13.1)

or

∏_{j=1}^{n} A_{j}. (1.13.2)

This is the set of n-tuples a = (a_{1}, …, a_{n}) such that a_{j} ∈ A_{j} for each j = 1, …, n.

Now let a_{1}, b_{1}, …, a_{n}, b_{n} be real numbers with a_{j} ≤ b_{j} for each j = 1, …, n, and put I_{j} = [a_{j}, b_{j}] for every j = 1, …, n. The Cartesian product

C = ∏_{j=1}^{n} I_{j} (1.13.3)

of I_{1}, …, I_{n} is called a cell in R^{n}. The diameter of C is defined by

diam C = ( ∑_{j=1}^{n}(b_{j} − a_{j})^{2})^{1/2}. (1.13.4)

This is the same as the maximum of the distances between elements of C, with respect to the standard Euclidean metric on R^{n}.

Proposition 1.13.5 If C_{1}, C_{2}, C_{3}, … is an infinite sequence of cells in R^{n} such that C_{l+1} ⊆ C_{l} (1.13.6)

for every positive integer l, then

⋂_{l=1}^{∞} C_{l} ≠ ∅. (1.13.7)

This is the same as Proposition 1.11.1 when n = 1, and we can reduce to that case otherwise. If l ∈ Z_{+}, then there are n closed intervals I_{1,l}, …, I_{n,l} in R such that

C_{l} = ∏_{j=1}^{n} I_{j,l}, (1.13.8)

as in (1.13.3). Using (1.13.6), we get that

I_{j,l+1} ⊆ I_{j,l} (1.13.9)

for every positive integer l and j = 1, …, n. This implies that

⋂_{l=1}^{∞} I_{j,l} ≠ ∅ (1.13.10)

for every j = 1, …, n, by Proposition 1.11.1. If x = (x_{1}, …, x_{n}) has the property that

x_{j} ∈ ⋂_{l=1}^{∞} I_{j,l} (1.13.11)

for each j = 1, …, n, then

x ∈ ⋂_{l=1}^{∞} C_{l}, (1.13.12)

as desired.

Proposition 1.13.13 If C is any cell in R^{n}, then C can be expressed as the union of 2^{n} cells, each of which has one-half the diameter of C.

As before, C can be expressed as in (1.13.3), where I_{j} = [a_{j}, b_{j}] is a closed interval in R for each j = 1, …, n. Let L_{j} = [a_{j}, (a_{j} + b_{j})/2] and R_{j} = [(a_{j} + b_{j})/2, b_{j}] be the left and right halves of I_{j} for each j = 1, …, n, so that I_{j} = L_{j} ∪ R_{j} for every j = 1, …, n. Consider the cells in R^{n} that can be obtained by taking the Cartesian product of n closed intervals, where for each j = 1, …, n, the jth interval is either L_{j} or R_{j}. It is easy to see that there are 2^{n} of these cells. One can check that the union of these 2^{n} cells is equal to C. One can also verify that the diameter of each of these 2^{n} cells is equal to diam C/2. This uses the fact that the lengths of L_{j} and R_{j} are equal to length(I_{j})/2 for each j = 1, …, n.

1.14 Compactness of cells

Theorem 1.14.1 If C is a cell in R^{n} for some positive integer n, then C is compact, with respect to the standard Euclidean metric on R^{n}.

Of course, this is the same as Theorem 1.12.1 when n = 1, and we can use essentially the same argument here. Let {U_{α}}_{α∈A} be an arbitrary open covering of C in R^{n}, and suppose for the sake of a contradiction that C cannot be covered by finitely many U_{α}’s. Under these conditions, we can find an infinite sequence of cells C_{1}, C_{2}, C_{3}, … with the following properties. First,

(1.14.2) C_{l} ⊆ C_{l−1} for every l ≥ 1,

where we put C_{0} = C, for convenience. Second,

(1.14.3) diam C_{l} = diam C_{l−1}/2 for every l ≥ 1.

Third,

(1.14.4) for each l ∈ Z_{+}, C_{l} cannot be covered by finitely many U_{α}’s.

To see this, suppose that C_{l} has already been chosen in this way for some nonnegative integer l. Using Proposition 1.13.13, we can express C_{l} as the union of 2^{n} cells, each of which has one-half the diameter of C_{l}. At least one of these 2^{n} cells cannot be covered by finitely many U_{α}’s, because otherwise C_{l} could be covered by finitely many U_{α}’s. We take C_{l+1} to be one of these 2^{n} cells, in such a way that C_{l+1} cannot be covered by finitely many U_{α}’s.

It follows that ⋂_{l=1}^{∞} C_{l} ≠ ∅, by Proposition 1.13.5. Let x be an element of ⋂_{l=1}^{∞} C_{l}, which is an element of C in particular. Thus there is an α_{0} ∈ A such that x ∈ U_{α_{0}}. Because U_{α_{0}} is an open set in R^{n}, there is a positive real number r such that B(x, r) ⊆ U_{α_{0}}. (1.14.5)

Here B(x, r) is the open ball in R^{n} centered at x with radius r with respect to the standard Euclidean metric. Let l be a positive integer that is large enough so that (1.14.6) diam C_{l} = 2^{−l} diam C < r.

It is easy to see that C_{l} ⊆ B(x, r) ⊆ U_{α_{0}}, (1.14.7)

because x ∈ C_{l}. This contradicts (1.14.4), as desired.

Corollary 1.14.8 If E ⊆ R^{n} is closed and bounded, with respect to the standard Euclidean metric, then E is compact.

Indeed, one can find a cell C ⊆ R^{n} such that E ⊆ C, because E is bounded in R^{n}. To get that E is compact, one can use the compactness of C and the hypothesis that E be a closed set, as in Proposition 1.9.6.

1.15 Totally bounded sets

Let (M, d(x, y)) be a metric space.

Definition 1.15.1 A subset E of M is said to be totally bounded with respect to d(·, ·) if for every positive real number r there are finitely many elements x_{1}, …, x_{n} of M such that

E ⊆ ⋃_{j=1}^{n} B(x_{j}, r). (1.15.2)

If E = ∅, then this condition is interpreted as holding with n = 0, even if M = ∅.

Proposition 1.15.3 If K ⊆ M is compact, then K is totally bounded.

Let r > 0 be given. The collection of open balls B(x, r) centered at elements x of K and with radius r forms an open covering of K in M. If K is compact, then there are finitely many elements x_{1}, …, x_{n} of K such that K ⊆ ⋃_{j=1}^{n} B(x_{j}, r), as desired.

Proposition 1.15.4 If E ⊆ M has the limit point property, then E is totally bounded in M.

Let r > 0 be given again. We would like to show that E is contained in the union of finitely many open balls in M of radius r. Of course, this is trivial when E = ∅, and so we may suppose that there is an element x_{1} of E. If

E ⊆ B(x_{1}, r), (1.15.5)

then we can stop. Otherwise, we let x_{2} be an element of E \ B(x_{1}, r).

Suppose that x_{1}, …, x_{n} ∈ E have been chosen in this way for some positive integer n. If E ⊆ ⋃_{j=1}^{n} B(x_{j}, r), then we can stop. Otherwise, we let x_{n+1} be an element of

E \ ( ⋃_{j=1}^{n} B(x_{j}, r)). (1.15.6)

Suppose for the sake of a contradiction that this process does not stop after finitely many steps. In this case, we get an infinite sequence x_{1}, x_{2}, x_{3}, … of elements of E such that d(x_{j}, x_{l}) ≥ r (1.15.7)

for all positive integers j, l with j < l. Let

L = {x_{j} : j ∈ Z_{+}} (1.15.8)

be the set of points that have been chosen in this way. Note that L has infinitely many elements, because x_{j} ≠ x_{l} when j < l. Of course, L ⊆ E, by construction.

If E has the limit point property, then there is a point p ∈ E such that p is a limit point of L in M. This implies that B(p, r/2) contains infinitely many elements of L, as in Proposition 1.5.2. In particular, there are positive integers j, l such that j < l and x_{j}, x_{l} ∈ B(p, r/2). It follows that

d(x_{j}, x_{l}) ≤ d(x_{j}, p) + d(p, x_{l}) < r/2 + r/2 = r, (1.15.9)

using the triangle inequality in the first step. This contradicts (1.15.7), as desired.

Metric spaces, continued

2.1 Countable sets

To say that a nonempty set A has finitely many elements means that there is a finite sequence x_{1}, …, x_{n} of elements of A in which every element of A occurs exactly once. In this case, the number of elements of A is the positive integer n. The empty set corresponds to taking n = 0.

Definition 2.1.1 A set A is said to be countably infinite if there is a sequence {x_{j}}_{j=1}^{∞} of elements of A in which every element of A occurs exactly once. More precisely, this means that x_{j} ∈ A for every j ∈ Z_{+}, and for every a ∈ A, there is a unique positive integer j such that x_{j} = a.

Of course, the set Z_{+} of positive integers is countably infinite, since we can take x_{j} = j for every j ≥ 1. The set Z of all integers is countably infinite as well. To see this, one can take

(2.1.2)  x_{j} = j/2  when j is even
= −(j − 1)/2 when j is odd.

Proposition 2.1.3 Let A and B be sets, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of B in which every element of A occurs at least once. More precisely, this means that x_{j} ∈ B for every j ≥ 1, and for every a ∈ A there is a positive integer j such that x_{j} = a. Under these conditions, A has only finitely or countably many elements.

Of course, if A = ∅, then A has only finitely many elements. Otherwise, let j_{1} be the smallest positive integer such that x_{j_{1}} ∈ A. If this is the only element of A, then we can stop. Otherwise, let j_{2} be the smallest positive integer such that x_{j_{2}} ∈ A and x_{j_{2}} ≠ x_{j_{1}}. Note that j_{1} < j_{2}, by construction.

Suppose that positive integers j_{1} < ⋯ < j_{n} have been chosen in this way for some n ∈ Z_{+}. If x_{j_{1}}, …, x_{j_{n}} are all of the elements of A, then we can stop. Otherwise, let j_{n+1} be the smallest positive integer such that x_{j_{n+1}} ∈ A and x_{j_{n+1}} ≠ x_{j_{l}} for each l = 1, …, n. By construction, j_{n} < j_{n+1}, since otherwise j_{n+1} would have been chosen earlier in the process.

If this process stops after finitely many steps, then A has only finitely many elements. Otherwise, we get an infinite sequence {x_{j_{l}}}_{l=1}^{∞} of elements of A in which every element of A occurs exactly once, so that A is countably infinite.

Corollary 2.1.4 If A is a subset of a countably infinite set B, then A has only finitely or countably many elements.

Lemma 2.1.5 If A_{1}, A_{2}, A_{3}, … is an infinite sequence of finite sets, then their union ⋃_{j=1}^{∞} A_{j} has only finitely or countably many elements.

In this situation, it is not difficult to find a sequence in which every element of ⋃_{j=1}^{∞} A_{j} occurs at least once, by listing the elements of A_{j} for each j ∈ Z_{+}, one after the other. To get that ⋃_{j=1}^{∞} A_{j} has only finitely of countably many elements, one can use Proposition 2.1.3.

Let us use the lemma to show that the set Z_{+}^{2} = Z_{+} × Z_{+} of all ordered pairs of positive integers is countably infinite. Put

A_{n} = {(j, l) ∈ Z_{+}^{2} : j + l = n + 1} (2.1.6)

for each positive integer n. It is easy to see that A_{n} has exactly n elements for every n, and that

⋃_{n=1}^{∞} A_{n} = Z_{+}^{2}. (2.1.7)

It follows that Z_{+}^{2} is countably infinite, by Lemma 2.1.5, and because Z_{+}^{2} obviously has infinitely many elements. One can also use the fact that the A_{n}’s are pairwise disjoint, to enumerate the elements of Z_{+}^{2} more directly.

Proposition 2.1.8 Let A_{1}, A_{2}, A_{3}, … be an infinite sequence of sets, each of which has only finitely or countably many elements. Under these conditions, ⋃_{j=1}^{∞} A_{j} has only finitely or countably many elements as well.

By hypothesis, for each positive integer j, there is an infinite sequence {x_{j,l}}_{l=1}^{∞} in which every element of A_{j} occurs at least once. Let {(j_{n}, l_{n})}_{n=1}^{∞} be a sequence of elements of Z_{+}^{2} in which every element of Z_{+}^{2} occurs once. It follows that every element of ⋃_{j=1}^{∞} A_{j} occurs at least once in the sequence {x_{j_{n},l_{n}}}_{n=1}^{∞}. This implies that ⋃_{j=1}^{∞} A_{j} has only finitely or countably many elements, as in Proposition 2.1.3.

Let us now use the proposition to show that the set Q of rational numbers is countably infinite. If j is a positive integer, then let A_{j} be the set of integer multiples of 1/j. It is easy to see that A_{j} is countably infinite for each j ≥ 1, because the set Z of all integers is countably infinite. It follows that Q = ⋃_{j=1}^{∞} A_{j} is countably infinite, as desired.

Proposition 2.1.9 Let n be a positive integer, and let A_{1}, …, A_{n} be n countably infinite sets. Under these conditions, ∏_{j=1}^{n} A_{j} is countably infinite too.

Of course, this is trivial when n = 1, and the n = 2 case corresponds to the countability of Z_{+}^{2}. If n ≥ 2, then there is a simple one-to-one correspondence

between ∏_{j=1}^{n} A_{j} and ( ∏_{j=1}^{n−1} A_{j}) × A_{n}. More precisely, if a = (a_{1}, …, a_{n}) is an element of ∏_{j=1}^{n} A_{j}, then

((a_{1}, …, a_{n−1}), a_{n}) (2.1.10)

defines an element of ( ∏_{j=1}^{n−1} A_{j}) × A_{n}, and every element of ( ∏_{j=1}^{n−1} A_{j}) × A_{n} corresponds to a unique element of ∏_{j=1}^{n} A_{j} in this way. Thus the countability of ∏_{j=1}^{n} A_{j} can be obtained from the countability of ∏_{j=1}^{n−1} A_{j} and the n = 2 case. This permits one to prove the proposition by induction.

2.2 Uncountable sets

Definition 2.2.1 A set is said to be uncountable if it is neither finite nor countably infinite.

Let B be the set of all infinite sequences x = {x_{j}}_{j=1}^{∞} such that for each j ∈ Z_{+}, x_{j} = 0 or 1. It is well known that B is uncountable. To see this, let x(1), x(2), x(3), … be any infinite sequence of elements of B. Thus, for each l ∈ Z_{+}, x(l) = {x_{j}(l)}_{j=1}^{∞} is an infinite sequence with terms in {0, 1}. Put

y_{j} = 1 − x_{j}(j) ∈ {0, 1} (2.2.2)

for each j ∈ Z_{+}, so that y = {y_{j}}_{j=1}^{∞} defines an element of B. If l is any positive integer, then y ≠ x(l), because y_{l} ≠ x_{l}(l), by construction. This shows that the elements of B cannot all be listed by a sequence, so that B is uncountable. This is Cantor’s diagonalization argument.

There is a standard way to associate to each x ∈ B an element of the closed unit interval [0, 1]. It is well known that every element of [0, 1] corresponds to an element of B in this way, but some elements of [0, 1] may correspond to more than one element of B. More precisely, it is well known and not too difficult to show that at most two elements of B can correspond to the same element of [0, 1]. In fact, this only happens for countably many elements of B, and hence countably many elements of [0, 1]. One can use this to show that [0, 1] is uncountable, because B is uncountable.

Definition 2.2.3 Let (M, d(x, y)) be a metric space. A subset E of M is said to be perfect in M if E is a closed set, and if every element of E is a limit point of E.

Closed intervals in the real line of positive length are perfect, for instance, with respect to the standard Euclidean metric on R. If E is a nonempty perfect subset of R^{n} for some positive integer n, with respect to the standard Euclidean metric, then it can be shown that E is uncountable. This can be extended to a broad class of metric spaces using the Baire category theorem. Another type of example of a perfect set in R will be discussed in Section 2.9.

2.3 Separable metric spaces

Let (M, d(x, y)) be a metric space.

Definition 2.3.1 A subset E of M is said to be dense in M if every element of M is either an element of E, a limit point of E, or both.

Equivalently, E ⊆ M is dense in M if the closure \overline{E} of E in M is equal to M.

Definition 2.3.2 If there is a dense set E ⊆ M such that E has only finitely or countably many elements, then M is said to be separable as a metric space.

It is easy to see that the set Q of rational numbers is dense in the real line, with respect to the standard Euclidean metric. Thus R is separable, with respect to the standard metric, because Q is countably infinite, as in Section 2.1.

Similarly, if n is a positive integer, then let Q^{n} be the set of x ∈ R^{n} such that x_{j} ∈ Q for each j = 1, …, n. One can check that Q^{n} is dense in R^{n}, with respect to the standard Euclidean metric. It follows from Proposition 2.1.9 that Q^{n} is countably infinite, so that R^{n} is separable with respect to the standard metric.

Proposition 2.3.3 Separability of M is equivalent to the following condition: for every r > 0 there is a subset E(r) of M such that E(r) has only finitely or countably many elements and

⋃_{x∈E(r)} B(x, r) = M. (2.3.4)

If M is separable, then there is a dense set E ⊆ M with only finitely or countably many elements. In this case, one can check that (2.3.4) holds with E(r) = E for every r > 0.

Conversely, suppose that M satisfies the condition described in the statement of the proposition. This implies that for each positive integer j there is a subset E(1/j) of M with only finitely or countably many elements such that (2.3.4) holds with r = 1/j. Put

E = ⋃_{j=1}^{∞} E(1/j), (2.3.5)

which has only finitely or countably many elements, by Proposition 2.1.8. One can check that E is dense in M, using (2.3.4) for r = 1/j, j ∈ Z_{+}. This means that M is separable, as desired.

Corollary 2.3.6 If M is totally bounded, then M is separable.

Indeed, if M is totally bounded, then for each r > 0 there is a finite subset E(r) of M that satisfies (2.3.4).

2.4 Bases

The topics in this section are a bit abstract, and one may wish to skip this at first. We shall say more about this in the next section.

Let (M, d(x, y)) be a metric space.

Definition 2.4.1 A collection B of open subsets of M is said to be a base for the topology of M if for every x ∈ M and r > 0 there is a V ∈ B such that x ∈ V and V ⊆ B(x, r).

If B is a base for the topology of M and W ⊆ M is an open set, then

W = ⋃{V : V ∈ B, V ⊆ W }. (2.4.2)

More precisely, the right side of (2.4.2) is automatically contained in W, by construction. In order to verify the opposite inclusion, let an arbitrary element x of W be given. Thus there is an r > 0 such that B(x, r) ⊆ W, because W is an open set in M. This implies that there is a V ∈ B such that x ∈ V and V ⊆ B(x, r) ⊆ W, as in Definition 2.4.1. It follows that x is an element of the right side of (2.4.2). This means that W is contained in the right side of (2.4.2), as desired.

Conversely, let B be a collection of open subsets of M, and suppose that every open subset of M can be expressed as the union of some elements of B. Let x ∈ M and r > 0 be given, and remember that B(x, r) is an open set in M. By hypothesis, B(x, r) can be expressed as a union of elements of B. In particular, there is a V ∈ B such that x ∈ V and V ⊆ B(x, r), because x ∈ B(x, r). This shows that B is a base for the topology of M.

Of course, the collection of all open subsets of M is a base for the topology of M. The collection of all open balls in M is a base for the topology of M as well, because of the way that open sets in M are defined. The collection of open balls in M with radius of the form 1/j for some positive integer j is a base for the topology of M too, because of the archimedean property of the real numbers.

Proposition 2.4.3 Let E be a subset of M, and let B_{E} be the collection of open balls in M of the form B(x, 1/j), with x ∈ E and j ∈ Z_{+}. If E is dense in M, then B_{E} is a base for the topology of M.

It is easy to see that E has to be dense in M in order for B_{E} to be a base for the topology of M.

Suppose that E is dense in M, and let us check that B_{E} is a base for the topology of M. Let x ∈ M and r > 0 be given, as in Definition 2.4.1. The archimedean property for the real numbers implies that there is a positive integer j such that 2/j ≤ r. Because E is dense in M, there is a y ∈ E such that d(x, y) < 1/j. Thus B(y, 1/j) ∈ B_{E} and x ∈ B(y, 1/j). One can also verify that

B(y, 1/j) ⊆ B(x, r), (2.4.4) using the triangle inequality. More precisely, if z ∈ B(y, 1/j), then

d(x, z) ≤ d(x, y) + d(y, z) < 1/j + 1/j = 2/j ≤ r, (2.4.5)

so that z ∈ B(x, r), as desired.

Corollary 2.4.6 If M is separable, then there is a base for the topology of M with only finitely or countably many elements.

Suppose that M is separable, and let E be a dense subset of M with only finitely or countably many elements. It suffices to check that the collection B_{E} defined in Proposition 2.4.3 has only finitely or countably many elements. If j is a positive integer, then let B_{E,j} be the collection of open balls of the form B(x, 1/j), with x ∈ E. Observe that

B_{E} = ⋃_{j=1}^{∞} B_{E,j}, (2.4.7)

by the definition of B_{E}. It is easy to see that B_{E,j} has only finitely or countably many elements for each j ∈ Z_{+}, because E has only finitely or countably many elements, by hypothesis. More precisely, one can list the elements of E with a finite or infinite sequence, and use that to list the elements of B_{E,j} with a finite or infinite sequence. This implies that (2.4.7) has only finitely or countably many elements, by Proposition 2.1.8.

Proposition 2.4.8 If there is a base B for the topology of M such that B has only finitely or countably many elements, then M is separable.

Let B be a base for the topology of M. If V ∈ B and V ≠ ∅, then let us pick a point x_{V} in V. Let E = {x_{V} : V ∈ B, V ≠ ∅} (2.4.9)

be the set of points in M that have been chosen in this way. One can check that E is dense in M, because B is a base for the topology of M. If B has only finitely or countably many elements, then E has only finitely or countably many elements. More precisely, if one can list the elements of B with a finite or infinite sequence, then one can list the elements of E with a finite or infinite sequence too. In this case, we get that M is separable, as desired.

2.5 Lindelöf’s theorem

The topics in this section are also a bit abstract, and one may wish to skip this at first. More precisely, the conclusion of the next theorem is quite interesting, and one may wish to skip the proof at first.

Let (M, d(x, y)) be a metric space.

Theorem 2.5.1 (Lindelöf’s theorem) Suppose that there is a base B for the topology of M with only finitely or countably many elements. If {U_{α}}_{α∈A} is any family of open subsets of M, then there is a subset A_{1} of A such that A_{1} has only finitely or countably many elements and

⋃_{α∈A_{1}} U_{α} = ⋃_{α∈A} U_{α}. (2.5.2)

Note that the hypothesis of Lindelöf’s theorem is the same as saying that M is separable as a metric space, as in the previous section. Lindelöf’s theorem will be used in the proof of Theorem 2.7.1.

To prove Lindelöf’s theorem, let α ∈ A be given, and put

B_{α} = {V ∈ B : V ⊆ U_{α}}. (2.5.3)

Note that

U_{α} = ⋃_{V ∈B_{α}} V, (2.5.4)

as in (2.4.2), because B is a base for the topology of M, and U_{α} is an open set in M. Put

B̃ = ⋃_{α∈A} B_{α}. (2.5.5)

It is easy to see that

⋃_{V ∈_{B}}V = ⋃_{α∈A} ⋃_{V ∈B_{α}} V = ⋃_{α∈A} U_{α}, (2.5.6)

using the definition (2.5.5) of B̃ in the first step, and (2.5.4) in the second step. If V ∈ B̃, then let us pick α(V) ∈ A such that V ∈ B_{α(V)}, which means that

V ⊆ U_{α(V)}. (2.5.7)

Let A_{1} = {α(V) : V ∈ B̃} (2.5.8)

be the collection of elements of A that have been chosen in this way. Observe that B̃ has only finitely or countably many elements, because B has only finitely of countably many elements, by hypothesis, and B̃ ⊆ B, by construction. This implies that A_{1} has only finitely or countably many elements as well. More precisely, the elements of B̃ can be listed by a finite or infinite sequence, which can be used to list the elements of A_{1} by a finite or infinite sequence.

We also have that

⋃_{α∈A_{1}} U_{α} = ⋃_{V ∈_{B}}U_{α(V)} ⊇ ⋃_{V ∈_{B}}V = ⋃_{α∈A} U_{α}, (2.5.9)

using the definition (2.5.8) of A_{1} in the first step, (2.5.7) in the second step, and (2.5.6) in the third step. Of course,

⋃_{α∈A_{1}} U_{α} ⊆ ⋃_{α∈A} U_{α} (2.5.10)

automatically, because A_{1} ⊆ A. Combining (2.5.9) and (2.5.10), we get (2.5.2), as desired.

2.6 Countable open coverings

Let (M, d(x, y)) be a metric space.

Proposition 2.6.1 Suppose that E ⊆ M has the limit point property. If U_{1}, U_{2}, U_{3}, … is a sequence of open subsets of M such that

E ⊆ ⋃_{j=1}^{∞} U_{j}, (2.6.2)

then there is a positive integer n such that

E ⊆ ⋃_{j=1}^{n} U_{j}. (2.6.3)

Suppose for the sake of a contradiction that

E ⊈ ⋃_{j=1}^{n} U_{j} (2.6.4)

for every n ∈ Z_{+}. Let us choose, for each n ∈ Z_{+}, a point

x_{n} ∈ E \ ( ⋃_{j=1}^{n} U_{j}). (2.6.5)

Let L = {x_{n} : n ∈ Z_{+}} (2.6.6)

be the set of points in E that have been chosen in this way.

Let us check that L has infinitely many elements. Otherwise, if L has only finitely many elements, then there is an x ∈ E such that x = x_{n} for infinitely many n ∈ Z_{+}. Because x ∈ E, there is a positive integer j_{0} such that x ∈ U_{j_{0}}, by (2.6.2). It follows that x_{n} ≠ x when n ≥ j_{0}, by (2.6.5). This means that x_{n} = x for only finitely many n, which is a contradiction, as desired.

If E has the limit point property, then there is a point p ∈ E that is a limit point of L in M. Using (2.6.2), we get that there is a positive integer j_{1} such that p ∈ U_{j_{1}}. Because U_{j_{1}} is an open set in M, by hypothesis, there is a positive real number r such that B(p, r) ⊆ U_{j_{1}}. (2.6.7) We also have that B(p, r) contains infinitely many elements of L, because p is a limit point of L, as in Proposition 1.5.2. This implies that x_{n} ∈ B(p, r) for infinitely many n ∈ Z_{+}.

It follows that x_{n} ∈ U_{j_{1}} (2.6.8) for infinitely many n ∈ Z_{+}, by (2.6.7). However, (2.6.8) can only hold when n < j_{1}, by (2.6.5). This is a contradiction, as desired.

2.7 Getting compactness

The proof of the next theorem uses Lindelöf’s theorem, as well as other results discussed previously. One should not consider the next theorem as being completely established here unless one has gone through the proof of Lindelöf’s theorem, as well as that of Corollary 2.4.6.

Let (M, d(x, y)) be a metric space.

Theorem 2.7.1 If E ⊆ M has the limit point property, then E is compact.

Remember that the converse was given in Proposition 1.7.10.

Suppose first that M has the limit point property, and let us show that M is compact. Remember that M is totally bounded in this situation, as in Proposition 1.15.4. This implies that M is separable, as in Corollary 2.3.6. It follows that there is a base B for the topology of M with only finitely or countably many elements, as in Corollary 2.4.6.

Let {U_{α}}_{α∈A} be an arbitrary open covering of M, as a subset of itself. Thus ⋃_{α∈A} U_{α} = M. Lindelöf’s theorem implies that there is a subset A_{1} of A such that A_{1} has only finitely or countably many elements, and

⋃_{α∈A_{1}} U_{α} = M. (2.7.2)

If A_{1} has only finitely many elements, then we can stop. Otherwise, suppose that A_{1} is countably infinite. In this case, we can use Proposition 2.6.1 to find a finite subset A_{2} of A_{1} such that

M ⊆ ⋃_{α∈A_{2}} U_{α}. (2.7.3)

In both cases, we get a finite subcovering of M from {U_{α}}_{α∈A}, as desired.

Suppose now that E is a subset of M with the limit point property. Remember that E may be considered as a metric space as well, with respect to the restriction of d(x, y) to x, y ∈ E. It is easy to see that E has the limit point property as a subset of itself, because of the corresponding property of E in M. Using this, the previous argument implies that E is compact as a subset of itself. It follows that E is compact as a subset of M, as in Proposition 1.10.12.

2.8 Connectedness

Let (M, d(x, y)) be a metric space.

Definition 2.8.1 A pair A, B of subsets of M are said to be separated in M if \overline{A} ∩ B = A ∩ \overline{B} = ∅. (2.8.2)

If we take A = (0, 1) and B = (1, 2) in the real line, then their closures with respect to the standard Euclidean metric on R are given by \overline{A} = [0, 1] and \overline{B} = [1, 2]. It follows that A and B are separated in R, although their closures are not disjoint in R.

Definition 2.8.3 A subset E of M is said to be connected in M if E cannot be expressed as the union of two nonempty separated subsets of M.

Theorem 2.8.4 A subset E of the real line is connected with respect to the standard metric on R if and only if it has the following property:

(2.8.5) if x, y ∈ E, w ∈ R, and x < w < y, then w ∈ E.

Suppose first that (2.8.5) does not hold, and let us show that E is not connected in R. By hypothesis, there are x, y ∈ E and w ∈ R such that x < w < y and w ∉ E. Put

A = {u ∈ E : u < w} and B = {u ∈ E : w < u}. (2.8.6)

Thus x ∈ A and y ∈ B, so that A, B ≠ ∅, and E = A ∪ B, because w ∉ E. It is easy to see that

\overline{A} ⊆ \{u ∈ R : u ≤ w\} and \overline{B} ⊆ \{u ∈ R : w ≤ u\}. (2.8.7)

This implies that (2.8.2) holds, so that A and B are separated in R. It follows that E is not connected in R, as desired.

Before proceeding to the other half of the theorem, let us mention the following useful fact.

Proposition 2.8.8 If A is a nonempty subset of R with an upper bound in R, then sup A ∈ \overline{A}. Similarly, if B is a nonempty subset of R with a lower bound in R, then inf B ∈ \overline{B}.

This can be verified directly from the definitions, and the details are left as an exercise.

Let us now prove the “if” part of Theorem 2.8.4. Thus we suppose that (2.8.5) holds, and we would like to show that E is connected in R. Suppose for the sake of a contradiction that E is not connected, so that there are nonempty separated sets A, B ⊆ R such that E = A ∪ B. Let x be an element of A, and let y be an element of B. We may as well suppose that x < y, since otherwise we could interchange the roles of A and B.

Note that A ∩ [x, y] is a nonempty subset of R with an upper bound in R, and put z = sup(A ∩ [x, y]). (2.8.9)

Clearly x ≤ z ≤ y, because x ∈ A ∩ [x, y], and y is an upper bound for A ∩ [x, y]. Using Proposition 2.8.8, we get that z is in the closure of A ∩ [x, y] in R. It is easy to see that this implies that z ∈ \overline{A}. It follows that z ∉ B, because \overline{A} ∩ B = ∅, by hypothesis.

Using (2.8.5), we get that z ∈ E. This implies that z ∈ A, because z ∉ B. Of course, z < y, because z ≤ y and z ≠ y, since y ∈ B and z ∉ B. We can use (2.8.5) again to get that (z, y) ⊆ E, because x ≤ z. It follows that

(z, y) ⊆ B, (2.8.10)

because E = A ∪ B, and (z, y) is disjoint from A, by the definition (2.8.9) of z. This means that z ∈ \overline{B}. This contradicts the hypothesis that A ∩ \overline{B} = ∅, as desired.

2.9 The Cantor set

Put E_{0} = [0, 1] and E_{1} = [0, 1/3]∪[2/3, 1]. Equivalently, E_{1} is obtained from E_{0} be removing the open middle third (1/3, 2/3). Similarly, we can remove the open middle thirds (1/9, 2/9) and (7/9, 8/9) from [0, 1/3] and [2/3, 1], respectively, to get E_{2} = [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [8/9, 1]. (2.9.1)

Continuing in this way, we can define a subset E_{j} of R for every nonnegative integer j with the following properties. First, for each j ≥ 0, E_{j} is the union of 2^{j} pairwise-disjoint closed intervals of length 3^{−j}. Second, E_{j+1} is obtained from E_{j} by removing the open middle thirds of each of the 2^{j} closed intervals in E_{j} just mentioned. In particular,

E_{j+1} ⊆ E_{j} (2.9.2)

for every j ≥ 0.

The middle-thirds Cantor set is defined to be

E = ⋂_{j=0}^{∞} E_{j}. (2.9.3)

Note that E_{j} is a closed set in R with respect to the standard Euclidean metric for every j ≥ 0, because it is the union of finitely many closed sets. It follows that E is a closed set in R too.

If j is a nonnegative integer, then let A_{j} be the set of 2^{j+1} elements of [0, 1] that occur as endpoints of the 2^{j} closed intervals that make up E_{j}. Thus A_{0} = {0, 1}, A_{1} = {0, 1/3, 2/3, 1}, A_{2} = {0, 1/9, 2/9, 1/3, 2/3, 7/9, 8/9, 1}, etc. By construction, A_{j} ⊆ E_{j} (2.9.4) for every j ≥ 0. We also have that

A_{j} ⊆ A_{j+1} (2.9.5)

for every j ≥ 0. More precisely, A_{j+1} consists of the elements of A_{j}, together with the endpoints of the open middle thirds of the closed intervals that make up E_{j}.

Let us check that A_{j} ⊆ E_{l} (2.9.6)

for all nonnegative integers j, l. If j ≤ l, then

A_{j} ⊆ A_{l} ⊆ E_{l}, (2.9.7)

using (2.9.5) in the first step, and (2.9.4) in the second step. Similarly, if j ≥ l, then A_{j} ⊆ E_{j} ⊆ E_{l}, (2.9.8)

using (2.9.4) in the first step, and (2.9.2) in the second step. It follows that

A_{j} ⊆ E (2.9.9)

for every j ≥ 0.

Observe that

A = ⋃_{j=0}^{∞} A_{j} (2.9.10)

is countably infinite. Clearly A ⊆ E, (2.9.11)

by (2.9.9). Let us check that every element of E is a limit point of A, with respect to the standard metric on R.

If x ∈ E and j is a nonnegative integer, then x ∈ E_{j}, and hence x is contained in one of the 2^{j} closed intervals I that make up E_{j}. The endpoints of I are elements of A_{j} ⊆ A, at least one of which is different from x. The distance from x to the endpoints of I is less than or equal to the length of I, which is 3^{−j}. Using this, it is easy to see that x is a limit point of A, as desired.

In particular, every element of E is a limit point of E, because of (2.9.11). This shows that E is perfect with respect to the standard metric on R, because E is a closed set.

Sequences and series

3.1 Complex numbers

Every complex number z can be expressed in a unique way as z = x+y i, where x, y ∈ R and i^{2} = −1. In this case, x and y are known as the real and imaginary parts of z, respectively. The complex conjugate of z is the complex number \overline{z} = x − y i. (3.1.1) The absolute value or modulus of z is the nonnegative real number |z| = (x^{2} + y^{2})^{1/2}. (3.1.2) Note that the complex conjugate of \overline{z} is z, and that |\overline{z}| = |z|.

Addition and multiplication of real numbers can be extended to the set C of complex numbers in a standard way. If z, w ∈ C, then one can check that \overline{z + w} = \overline{z} + \overline{w} (3.1.3) and \overline{z w} = \overline{z} \overline{w}. (3.1.4) We also have that z \overline{z} = |z|^{2} (3.1.5) for every z ∈ C. It follows that |z w|^{2} = z w \overline{z w} = z \overline{z} w \overline{w} = |z|^{2} |w|^{2} (3.1.6) for every z, w ∈ C, so that |z w| = |z| |w|. (3.1.7) If z ∈ C and z ≠ 0, then |z| > 0, and z (\overline{z}/|z|^{2}) = 1, (3.1.8) which means that \overline{z}/|z|^{2} is the multiplicative inverse of z in C.

It is well known that |z + w| ≤ |z| + |w| (3.1.9)

for every z, w ∈ C, which can be verified directly. If we identify (x, y) ∈ R^{2} with z = x + y i ∈ C, then the absolute value of a complex number corresponds exactly to the standard Euclidean norm on R^{2}. Thus (3.1.9) is the same as the triangle inequality for the standard Euclidean norm on R^{2}. It is easy to see that d(z, w) = |z − w| (3.1.10)

defines a metric on C, which is the standard metric on C. This corresponds exactly to the standard Euclidean metric on R^{2}.

3.2 Convergent sequences

Let (M, d(x, y)) be a metric space.

Definition 3.2.1 A sequence {x_{j}}_{j=1}^{∞} of elements of M is said to converge to an element x of M if for every positive real number ϵ there is a positive integer L such that d(x_{j}, x) < ϵ (3.2.2)

for every j ≥ L.

Proposition 3.2.3 If {x_{j}}_{j=1}^{∞} is a sequence of elements of M that converges to x ∈ M and to x′ ∈ M, then x = x′.

Suppose for the sake of a contradiction that x ≠ x′, so that d(x, x′) > 0. Because {x_{j}}_{j=1}^{∞} converges to x in M, there is a positive integer L such that

d(x_{j}, x) < d(x, x′)/2 (3.2.4)

for every j ≥ L. Similarly, there is an L′ ∈ Z_{+} such that

d(x_{j}, x′) < d(x, x′)/2 (3.2.5)

for every j ≥ L′, because {x_{j}}_{j=1}^{∞} converges to x′ in M. If j ≥ max(L, L′), then it follows that

d(x, x′) ≤ d(x, x_{j}) + d(x_{j}, x′) < d(x, x′)/2 + d(x, x′)/2 = d(x, x′), (3.2.6)

using the triangle inequality in the first step. In particular, this holds when j = max(L, L′), which is a contradiction, as desired.

If {x_{j}}_{j=1}^{∞} is a sequence of elements of M that converges to an element x of M, then x is called the limit of {x_{j}}_{j=1}^{∞}, and we put

lim_{j→∞} x_{j} = x. (3.2.7)

We may also say that x_{j} tends to x as j → ∞ in this case, or x_{j} → x as j → ∞.

Definition 3.2.8 A sequence {x_{j}}_{j=1}^{∞} of elements of M is said to be bounded in M if the set {x_{j} : j ∈ Z_{+}} (3.2.9)

of its terms is bounded in M.

Proposition 3.2.10 If {x_{j}}_{j=1}^{∞} is a sequence of elements of M that converges to an element x of M, then {x_{j}}_{j=1}^{∞} is bounded in M.

Because {x_{j}}_{j=1}^{∞} converges to x in M, there is a positive integer L such that

d(x_{j}, x) < 1 (3.2.11)

for every j ≥ L. If L = 1, then it follows that the set of the terms in this sequence is contained in B(x, 1), as desired. Otherwise, put

r = max{d(x_{j}, x) : j = 1, …, L − 1}, (3.2.12)

which is a nonnegative real number. It is easy to see that

d(x_{j}, x) < r + 1 (3.2.13)

for every j ≥ 1, so that the set of terms in the sequence is contained in B(x, r+1), as desired.

Proposition 3.2.14 (a) Let {x_{j}}_{j=1}^{∞} be a sequence of real numbers that increases monotonically, in the sense that

x_{j} ≤ x_{j+1} (3.2.15)

for every j ≥ 1. If the set of x_{j}’s, j ∈ Z_{+}, has an upper bound in R, then {x_{j}}_{j=1}^{∞} converges to a real number x, with respect to the standard Euclidean metric on R.

(b) Let {y_{j}}_{j=1}^{∞} be a monotonically decreasing sequence of real numbers, so that y_{j+1} ≤ y_{j} (3.2.16)

for every j ≥ 1. If the set of y_{j}’s has a lower bound in R, then {y_{j}}_{j=1}^{∞} converges to a real number y, with respect to the standard metric on R.

To prove (a), put A = {x_{j} : j ∈ Z_{+}}, and x = sup A. Let ϵ > 0 be given, and note that x−ϵ is not an upper bound for A, by definition of the supremum. Thus there is a positive integer L such that

x − ϵ < x_{L}. (3.2.17)

Of course, x_{j} ≤ x for every j ≥ 1, because x is an upper bound for A. If j ≥ L, then x_{j} ≥ x_{L} > x − ϵ, (3.2.18)

because the x_{j}’s increase monotonically. It follows that

|x_{j} − x| = x − x_{j} < ϵ (3.2.19)

for every j ≥ L, as desired. Part (b) can be shown in essentially the same way, or by reducing to the previous case.

3.3 Sums and products

In this section, we consider some basic properties of convergent sequences of complex numbers. Of course, this uses the standard metric on C.

Proposition 3.3.1 Let {z_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞} be sequences of complex numbers that converge to complex numbers z and w, respectively. Under these conditions, {z_{j} + w_{j}}_{j=1}^{∞} converges to z + w in C.

Thus

(3.3.2)  lim_{j→∞}(z_{j} + w_{j}) = lim_{j→∞} z_{j} + lim_{j→∞} w_{j},

where more precisely the limits on the right exist by hypothesis, and the existence of the limit on the left is part of the conclusion. To see this, let ϵ > 0 be given. Because {z_{j}}_{j=1}^{∞} converges to z in C, there is a positive integer L_{1} such that |z_{j} − z| < ϵ/2 (3.3.3)

for every j ≥ L_{1}. Similarly, because {w_{j}}_{j=1}^{∞} converges to w in C, there is an L_{2} ∈ Z_{+} such that |w_{j} − w| < ϵ/2 (3.3.4)

for every j ≥ L_{2}. If j ≥ max(L_{1}, L_{2}), then it follows that

(3.3.5) |(z_{j} + w_{j}) − (z + w)| = |(z_{j} − z) + (w_{j} − w)|
≤ |z_{j} − z| + |w_{j} − w| < ϵ/2 + ϵ/2 = ϵ,

as desired.

Lemma 3.3.6 If {a_{j}}_{j=1}^{∞} is a sequence of complex numbers that converges to 0, and {b_{j}}_{j=1}^{∞} is a bounded sequence of complex numbers, then {a_{j} b_{j}}_{j=1}^{∞} converges to 0 in C.

The boundedness of {b_{j}}_{j=1}^{∞} in C implies that there is a positive real number B such that |b_{j}| ≤ B (3.3.7)

for every j ≥ 1. Let ϵ > 0 be given. Because {a_{j}}_{j=1}^{∞} converges to 0 in C, there is a positive integer L such that

|a_{j}| < ϵ/B (3.3.8)

for every j ≥ L. This implies that

|a_{j} b_{j}| = |a_{j}| |b_{j}| ≤ B |a_{j}| < B (ϵ/B) = ϵ (3.3.9)

for every j ≥ L, as desired.

Lemma 3.3.10 Let {z_{j}}_{j=1}^{∞} be a sequence of complex numbers that converges to a complex number z. If c ∈ C, then {c z_{j}}_{j=1}^{∞} converges to c z in C.

Equivalently, this says that

lim_{j→∞}(c z_{j}) = c ( lim_{j→∞} z_{j}), (3.3.11)

where the limit on the right exists by hypothesis, and the existence of the limit on the left is part of the conclusion.

Of course, if c = 0, then c z_{j} = 0 for every j ≥ 1, c z = 0, and the lemma is trivial. Thus we may suppose that c ≠ 0. Let ϵ > 0 be given. Because {z_{j}}_{j=1}^{∞} converges to z in C, there is a positive integer L such that

|z_{j} − z| < ϵ/|c| (3.3.12)

for every j ≥ L. This implies that

|c z_{j} − c z| = |c| |z_{j} − z| < |c| (ϵ/|c|) = ϵ (3.3.13)

for every j ≥ L, as desired.

Proposition 3.3.14 If {z_{j}}_{j=1}^{∞} and {w_{j}}_{j=1}^{∞} are sequences of complex numbers that converge to z, w ∈ C, respectively, then {z_{j} w_{j}}_{j=1}^{∞} converges to z w in C.

As before, this can be summarized by saying that

(3.3.15)  lim_{j→∞}(z_{j} w_{j}) = ( lim_{j→∞} z_{j}) ( lim_{j→∞} w_{j}),

where the limits on the right exist by hypothesis, and the existence of the limit on the left is part of the conclusion.

Observe that z_{j} w_{j} = z_{j} (w_{j} − w) + z_{j} w (3.3.16)

for every j ≥ 1. The hypothesis that {w_{j}}_{j=1}^{∞} converges to w in C is equivalent to saying that {w_{j} − w}_{j=1}^{∞} converges to 0. We also have that {z_{j}}_{j=1}^{∞} is a bounded sequence in C, because it converges, as in Proposition 3.2.10. It follows that {z_{j} (w_{j} − w)}_{j=1}^{∞} converges to 0 in C, as in Lemma 3.3.6. Using Lemma 3.3.10, we get that {z_{j} w}_{j=1}^{∞} converges to z w in C. Thus the right side of (3.3.16) is the sum of two convergent sequences in C. This implies that {z_{j} w_{j}}_{j=1}^{∞} converges to z w, by Proposition 3.3.1, as desired.

Proposition 3.3.17 If {z_{j}}_{j=1}^{∞} is a sequence of nonzero complex numbers that converges to a nonzero complex number z, then {1/z_{j}}_{j=1}^{∞} converges to 1/z in C.

This means that

lim_{j→∞}(1/z_{j}) = 1/( lim_{j→∞} z_{j}), (3.3.18)

where the existence of the limit on the right is part of the hypothesis, and the existence of the limit on the left is part of the conclusion.

3.4. SUBSEQUENCES AND SEQUENTIAL COMPACTNESS 39

Let us begin by observing that 1/z_{j} − 1/z = (z − z_{j})/(z_{j} z) (3.3.19) for every j ≥ 1. Thus |1/z_{j} − 1/z| = |z − z_{j}|/(|z_{j}| |z|) (3.3.20) for every j ≥ 1. We should first deal with the factor of |z_{j}| in the denominator.

Because {z_{j}}_{j=1}^{∞} converges to z and z ≠ 0, there is a positive integer L_{0} such that |z_{j} − z| < |z|/2 (3.3.21) for every j ≥ L_{0}. Using the triangle inequality, we get that |z| ≤ |z_{j}| + |z − z_{j}| < |z_{j}| + |z|/2 (3.3.22) when j ≥ L_{0}. This implies that |z|/2 < |z_{j}| (3.3.23) for every j ≥ L_{0}. Combining this with (3.3.20), we get that |1/z_{j} − 1/z| ≤ (2/|z|^{2}) |z − z_{j}| (3.3.24) for every j ≥ L_{0}.

Let ϵ > 0 be given. Because {z_{j}}_{j=1}^{∞} converges to z, there is a positive integer L such that |z_{j} − z| < (|z|^{2}/2) ϵ (3.3.25) for every j ≥ L. If j ≥ max(L_{0}, L), then it follows that |1/z_{j} − 1/z| < (2/|z|^{2}) (|z|^{2}/2) ϵ = ϵ, (3.3.26) as desired.

3.4 Subsequences and sequential compactness

Definition 3.4.1 Let {x_{j}}_{j=1}^{∞} be a sequence of elements of some set, and let {j_{l}}_{l=1}^{∞} be a strictly increasing sequence of positive integers, so that j_{l} < j_{l+1} for every l ≥ 1. Under these conditions, {x_{j_{l}}}_{l=1}^{∞} is called a subsequence of {x_{j}}_{j=1}^{∞}.

Note that a sequence may be considered as a subsequence of itself.

Let (M, d(x, y)) be a metric space. If {x_{j}}_{j=1}^{∞} is a sequence of elements of M that converges to an element x of M, and if {x_{j_{l}}}_{l=1}^{∞} is a subsequence of {x_{j}}_{j=1}^{∞}, then it is easy to see that {x_{j_{l}}}_{l=1}^{∞} converges to x in M as well. This uses the fact that j_{l} ≥ l for every l ≥ 1.

Definition 3.4.2 A subset K of M is said to be sequentially compact if for every sequence {x_{j}}_{j=1}^{∞} of elements of K there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} and an element x of K such that {x_{j_{l}}}_{l=1}^{∞} converges to x in M.

Proposition 3.4.3 A subset K of M is sequentially compact if and only if K has the limit point property in M.

Suppose first that K has the limit point property, and let us show that K is sequentially compact. Let {x_{j}}_{j=1}^{∞} be any sequence of elements in K, and let L = {x_{j} : j ∈ Z_{+}} be the set of terms in this sequence. If L has only finitely many elements, then there is an x ∈ K such that x_{j} = x for infinitely many j ∈ Z_{+}. Equivalently, this means that there is subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} such that x_{j_{l}} = x for every l ≥ 1. Of course, {x_{j_{l}}}_{l=1}^{∞} converges to x in this case.

Suppose now that L has infinitely many elements. If K has the limit point property, then there is a point x ∈ K that is a limit point of L in M. This implies that for every r > 0, there are infinitely many elements of L in B(x, r), as in Proposition 1.5.2. It follows that for each r > 0,

d(x, x_{j}) < r (3.4.4)

for infinitely many j ∈ Z_{+}.

Using this, we can find a subsequence {x_{j_{n}}}_{n=1}^{∞} of {x_{j}}_{j=1}^{∞} such that

d(x, x_{j_{n}}) < 1/n (3.4.5)

for every n ≥ 1. More precisely, we can first choose j_{1} ∈ Z_{+} so that (3.4.5) holds with n = 1. If j_{n} ∈ Z_{+} has been chosen for some positive integer n, then we can choose j_{n+1} ∈ Z_{+} so that j_{n+1} > j_{n} and (3.4.5) holds with n replaced with n + 1. It is easy to see that {x_{j_{n}}}_{n=1}^{∞} converges to x in M, because of (3.4.5).

Conversely, suppose that K ⊆ M is sequentially compact, and let us show that K has the limit point property. Let L be an infinite subset of K, and let {x_{j}}_{j=1}^{∞} be an infinite sequence of distinct elements of L. If K is sequentially compact, then there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} that converges to an element x of K in M. Let r > 0 be given, so that d(x_{j_{l}}, x) < r for all but finitely many l ∈ Z_{+}. This implies that there are infinitely many elements of L in B(x, r), so that x is a limit point of L in M, as desired.

3.5 Cauchy sequences and completeness

Let (M, d(x, y)) be a metric space.

Definition 3.5.1 A sequence {x_{j}}_{j=1}^{∞} of elements of M is said to be a Cauchy sequence if for every ϵ > 0 there is a positive integer L such that

d(x_{j}, x_{l}) < ϵ (3.5.2)

for every j, l ≥ L.

3.5. CAUCHY SEQUENCES AND COMPLETENESS 41

Proposition 3.5.3 If a sequence {x_{j}}_{j=1}^{∞} of elements of M converges to an element x of M, then {x_{j}}_{j=1}^{∞} is a Cauchy sequence in M.

Let ϵ > 0 be given. Because {x_{j}}_{j=1}^{∞} converges to x in M, there is a positive integer L such that d(x_{j}, x) < ϵ/2 (3.5.4) for every j ≥ L. If j, l ≥ L, then it follows that d(x_{j}, x_{l}) ≤ d(x_{j}, x) + d(x, x_{l}) < ϵ/2 + ϵ/2 = ϵ, (3.5.5) as desired. Proposition 3.5.6 If {x_{j}}_{j=1}^{∞} is a Cauchy sequence in M, then {x_{j}}_{j=1}^{∞} is bounded in M.

By hypothesis, there is a positive integer L such that d(x_{j}, x_{l}) < 1 (3.5.7) for every j, l ≥ L. If L = 1, then we get that x_{j} ∈ B(x_{1}, 1) for every j ≥ 1, as desired. Otherwise, put r = max{d(x_{j}, x_{L}) : j = 1, …, L − 1}, (3.5.8) which is a nonnegative real number. In this case, we have that x_{j} ∈ B(x_{L}, r +1) for every j ≥ 1, as desired. Proposition 3.5.9 Let {x_{j}}_{j=1}^{∞} be a Cauchy sequence of elements of M. If there is a subsequence {x_{j_{n}}}_{n=1}^{∞} of {x_{j}}_{j=1}^{∞} that converges to an element x of M, then {x_{j}}_{j=1}^{∞} converges to x in M.

Let ϵ > 0 be given. Because {x_{j}}_{j=1}^{∞} is a Cauchy sequence in M, there is a positive integer L such that d(x_{j}, x_{l}) < ϵ/2 (3.5.10) for every j, l ≥ L. Similarly, there is a positive integer N such that d(x_{j_{n}}, x) < ϵ/2 (3.5.11) for every n ≥ N, because {x_{j_{n}}}_{n=1}^{∞} converges to x in M. If l ≥ L, n ≥ N, and j_{n} ≥ L, then we get that d(x_{l}, x) ≤ d(x_{l}, x_{j_{n}}) + d(x_{j_{n}}, x) < ϵ/2 + ϵ/2 = ϵ. (3.5.12) As before, j_{n} ≥ n for every positive integer n, so that j_{n} ≥ L when n ≥ L. In particular, if n = max(L, N), then j_{n} ≥ L and n ≥ N. Using this, we obtain that d(x_{l}, x) < ϵ (3.5.13) for every l ≥ L, as desired.

Corollary 3.5.14 If K ⊆ M is sequentially compact, {x_{j}}_{j=1}^{∞} is a Cauchy sequence in M, and x_{j} ∈ K for every j ≥ 1, then {x_{j}}_{j=1}^{∞} converges to an element of K in M.

Definition 3.5.15 If every Cauchy sequence of elements of M converges to an element of M, then M is said to be complete as a metric space.

Corollary 3.5.16 If n is a positive integer, then R^{n} is complete with respect to the standard Euclidean metric.

Let {x_{j}}_{j=1}^{∞} be a Cauchy sequence of elements of R^{n}. It is easy to see that there is a cell C in R^{n} such that x_{j} ∈ C for every j ≥ 1, because {x_{j}}_{j=1}^{∞} is bounded in R^{n}, as in Proposition 3.5.6. Remember that C is compact in R^{n}, so that C has the limit point property and is sequentially compact, by Proposition 1.7.10, Theorem 1.14.1, and Proposition 3.4.3. Thus {x_{j}}_{j=1}^{∞} converges to an element of C, as in Corollary 3.5.14.

3.6 Some particular sequences

In this section, we consider the convergence of some particular sequences of real or complex numbers, with respect to the standard metrics on R and C.

Let p be a positive real number, so that a^{p} can be defined as a positive real number for every positive real number a, as in Section 1.3. It is well known that

lim_{j→∞} 1/j^{p} = 0. (3.6.1)

To see this, let ϵ > 0 be given. Observe that

1/j^{p} < ϵ (3.6.2)

if and only if j > (1/ϵ)^{1/p}. (3.6.3)

This holds for all but finitely many positive integers j, by the archimedean property of the real numbers. This is all a bit simpler when p = 1/k for some positive integer k. If p is any positive real number, then there is a positive integer k such that 1/k ≤ p, by the archimedean property, which can be used to reduce to that case.

If a is a nonnegative real number with a < 1, then

lim_{j→∞} a^{j} = 0. (3.6.4)

This is trivial when a = 0, and so we may as well suppose that a > 0. Put b = 1/a − 1 > 0. It is easy to see that

(1 + b)^{j} ≥ 1 + b j (3.6.5) for every positive integer j, using induction, for instance. This implies that

a^{j} = (1 + b)^{−j} ≤ 1/(1 + b j) (3.6.6)

for every j ≥ 1, and (3.6.4) follows easily from this.

Let α be a positive real number, and let us show that

lim_{j→∞} j^{α} a^{j} = 0. (3.6.7)

This follows from (3.6.6) when a > 0 and α < 1. Otherwise, the archimedean property for the real numbers implies that there is a positive integer k such that α < k. Note that a^{1/k} < 1, so that

lim_{j→∞} j^{α/k} a^{j/k} = 0, (3.6.8)

as before. This implies that

j^{α} a^{j} = (j^{α/k} a^{j/k})^{k} → 0 as j → ∞, (3.6.9)

as desired.

If p is a positive real number, then

lim_{j→∞} p^{1/j} = 1. (3.6.10)

Suppose first that p ≥ 1, so that p^{1/j} ≥ 1 for every j ∈ Z_{+}. Let ϵ > 0 be given, and observe that p^{1/j} < 1 + ϵ (3.6.11)

if and only if p < (1 + ϵ)^{j}. (3.6.12)

This holds for all but finitely many j ∈ Z_{+}, by (3.6.5), with b = ϵ. Thus (3.6.10) holds when p ≥ 1. If 0 < p ≤ 1, then we can apply the previous argument to 1/p, to get that

lim_{j→∞} 1/p^{1/j} = 1. (3.6.13)

This implies (3.6.10), by Proposition 3.3.17.

In fact,

lim_{j→∞} j^{1/j} = 1. (3.6.14)

Let ϵ > 0 be given, and note that j^{1/j} ≥ 1 for every j ∈ Z_{+}. As before,

j^{1/j} < 1 + ϵ (3.6.15)

if and only if j < (1 + ϵ)^{j}, (3.6.16)

which is the same as saying that

j (1 + ϵ)^{−j} < 1. (3.6.17)

The left side of (3.6.17) tends to 0 as j → ∞, as in (3.6.7). This implies that (3.6.17) holds for all but finitely many j ∈ Z_{+}, as desired.

If z is a complex number such that |z| < 1, then

lim_{j→∞} z^{j} = 0. (3.6.18)

More precisely, |z^{j}| = |z|^{j} → 0 as j → ∞, (3.6.19)

by (3.6.4).

3.7 Infinite series

Definition 3.7.1 Let a_{1}, a_{2}, a_{3}, … be an infinite sequence of complex numbers. The infinite series ∑_{j=1}^{∞} a_{j} is said to converge if the corresponding sequence of partial sums

s_{n} = ∑_{j=1}^{n} a_{j} (3.7.2)

converges as a sequence of complex numbers, with respect to the standard metric on C. In this case, we put

∑_{j=1}^{∞} a_{j} = lim_{n→∞} s_{n}. (3.7.3)

Sometimes we may wish to consider infinite series ∑_{j=0}^{∞} a_{j} starting at j = 0, or some other integer.

Proposition 3.7.4 Let ∑_{j=1}^{∞} a_{j} and ∑_{j=1}^{∞} b_{j} be convergent series of complex numbers.

(a) Under these conditions, ∑_{j=1}^{∞}(a_{j} + b_{j}) converges too, with

∑_{j=1}^{∞}(a_{j} + b_{j}) = ∑_{j=1}^{∞} a_{j} + ∑_{j=1}^{∞} b_{j}. (3.7.5)

(b) If c is a complex number, then ∑_{j=1}^{∞} c a_{j} converges, with

∑_{j=1}^{∞} c a_{j} = c ∑_{j=1}^{∞} a_{j}. (3.7.6)

If n is any positive integer, then

∑_{j=1}^{n}(a_{j} + b_{j}) = ∑_{j=1}^{n} a_{j} + ∑_{j=1}^{n} b_{j} (3.7.7)

and

∑_{j=1}^{n} c a_{j} = c ∑_{j=1}^{n} a_{j}. (3.7.8)

Thus Proposition 3.7.4 follows from the analogous statements for sequences of complex numbers.

Proposition 3.7.9 An infinite series ∑_{j=1}^{∞} a_{j} of complex numbers converges if and only if the following condition holds: for every ϵ > 0 there is a positive integer L such that

|∑_{j=l}^{n} a_{j}| < ϵ (3.7.10)

for all positive integers l, n with n ≥ l ≥ L.

One can check that this condition is equivalent to saying that the corresponding sequence of partial sums (3.7.2) is a Cauchy sequence with respect to the standard metric on C. Thus the proposition follows from the completeness of C with respect to the standard metric, which is the same as the completeness of R^{2} with respect to the standard Euclidean metric.

Corollary 3.7.11 If ∑_{j=1}^{∞} a_{j} is a convergent series of complex numbers, then {a_{j}}_{j=1}^{∞} converges to 0 as a sequence of complex numbers.

This follows by taking l = n in (3.7.10).

Definition 3.7.12 An infinite series ∑_{j=1}^{∞} a_{j} of complex numbers is said to converge absolutely if ∑_{j=1}^{∞} |a_{j}| converges as an infinite series of nonnegative real numbers.

Proposition 3.7.13 If ∑_{j=1}^{∞} a_{j} is an absolutely convergent series of complex numbers, then ∑_{j=1}^{∞} a_{j} converges.

Observe that

|∑_{j=l}^{n} a_{j}| ≤ ∑_{j=l}^{n} |a_{j}| (3.7.14)

for all positive integers l, n with l ≤ n, by the triangle inequality. One can use this to obtain Proposition 3.7.13 from Proposition 3.7.9.

Proposition 3.7.15 Let ∑_{j=1}^{∞} a_{j} be an infinite series of nonnegative real numbers. Under these conditions, ∑_{j=1}^{∞} a_{j} converges if and only if the corresponding sequence of partial sums (3.7.2) is bounded.

If ∑_{j=1}^{∞} a_{j} is a convergent series of complex numbers, then the corresponding sequence of partial sums is bounded, as in Proposition 3.2.10. If a_{j} is a nonnegative real number for each j ≥ 1, then it is easy to see that the partial sums are monotonically increasing. In this case, the boundedness of the partial sums implies that they converge as a sequence of real numbers, as in Proposition 3.2.14.

Proposition 3.7.16 (Comparison test) Let ∑_{j=1}^{∞} a_{j} be an infinite series of complex numbers, and let ∑_{j=1}^{∞} b_{j} be an infinite series of nonnegative real numbers. If |a_{j}| ≤ b_{j} for every j ≥ 1 and ∑_{j=1}^{∞} b_{j} converges, then ∑_{j=1}^{∞} a_{j} converges absolutely.

Observe that

∑_{j=1}^{n} |a_{j}| ≤ ∑_{j=1}^{n} b_{j} (3.7.17)

for every positive integer n. If ∑_{j=1}^{∞} b_{j} converges, then its sequence of partial sums is bounded. This implies that the partial sums of ∑_{j=1}^{∞} |a_{j}| are bounded as well, so that this series converges too.

Let z be a complex number, and let us consider the corresponding geometric series ∑_{j=0}^{∞} z^{j}. Here z^{j} is interpreted as being equal to 1 when j = 0. If n is a nonnegative integer, then

(1 − z) ∑_{j=0}^{n} z^{j} = ∑_{j=0}^{n} z^{j} − ∑_{j=0}^{n} z^{j+1} = ∑_{j=0}^{n} z^{j} − ∑_{j=1}^{n+1} z^{j} = 1 − z^{n+1}. (3.7.18)

This implies that

∑_{j=0}^{n} z^{j} = (1 − z^{n+1})/(1 − z). (3.7.19)

when z ≠ 1. If |z| < 1, then we get that ∑_{j=0}^{∞} z^{j} converges, with

∑_{j=0}^{∞} z^{j} = 1/(1 − z), (3.7.20)

by (3.6.18). Note that ∑_{j=0}^{∞} z^{j} converges absolutely in this case, by the same argument for |z|. If |z| ≥ 1, then

|z^{j}| = |z|^{j} ≥ 1 (3.7.21)

for every j ≥ 0. This means that {z^{j}}_{j=0}^{∞} does not converge to 0, so that ∑_{j=0}^{∞} z^{j} does not converge.

3.8 Power series

Let a_{0}, a_{1}, a_{2}, a_{3}, … be an infinite sequence of complex numbers, and consider the corresponding power series ∑_{j=0}^{∞} a_{j} z^{j}, for z ∈ C.

Proposition 3.8.1 If ∑_{j=0}^{∞} a_{j} z^{j} converges absolutely for some z ∈ C, then ∑_{j=0}^{∞} a_{j} w^{j} converges absolutely for every w ∈ C with |w| ≤ |z|.

Indeed, if |w| ≤ |z|, then

|a_{j} w_{j}| = |a_{j}| |w|^{j} ≤ |a_{j}| |z|^{j} = |a_{j} z^{j}| (3.8.2)

for every nonnegative integer j. Thus Proposition 3.8.1 follows from the comparison test.

Proposition 3.8.3 If ∑_{j=0}^{∞} a_{j} z^{j} converges for some nonzero complex number z, then ∑_{j=0}^{∞} a_{j} w^{j} converges absolutely for every w ∈ C with |w| < |z|.

If ∑_{j=0}^{∞} a_{j} z^{j} converges, then {a_{j} z^{j}}_{j=0}^{∞} converges to 0, by Corollary 3.7.11. This implies that {a_{j} z^{j}}_{j=0}^{∞} is a bounded sequence in C, as in Proposition 3.2.10. It follows that there is a nonnegative real number C such that

|a_{j} z^{j}| ≤ C (3.8.4)

for every j ≥ 0. Let w ∈ C be given, and observe that

|a_{j} w^{j}| = |a_{j} z^{j}| (|w|^{j}/|z|^{j}) ≤ C (|w|/|z|)^{j} (3.8.5)

for every j ≥ 0, by (3.8.4). If |w| < |z|, then ∑_{j=0}^{∞}(|w|/|z|)^{j} is a convergent geometric series, as in the previous section. Of course, this implies that ∑_{j=0}^{∞} C (|w|/|z|)^{j} converges too, as in Proposition 3.7.4. Proposition 3.8.3 now follows from the comparison test.

Let E be the set of complex numbers z such that ∑_{j=0}^{∞} a_{j} z^{j} does not converge, and put E_{1} = {|z| : z ∈ E}. (3.8.6) This is a set of nonnegative real numbers, which is not empty when E ≠ ∅.

Definition 3.8.7 The radius of convergence ρ of ∑_{j=0}^{∞} a_{j} z^{j} is defined by

ρ = inf E_{1} (3.8.8)

when E ≠ ∅, and by ρ = +∞ when E = ∅.

Proposition 3.8.9 (a) If w ∈ C satisfies |w| > ρ, then ∑_{j=0}^{∞} a_{j} w^{j} does not converge.

(b) If w ∈ C satisfies |w| < ρ, then ∑_{j=0}^{∞} a_{j} w^{j} converges absolutely.

(c) The radius of convergence of ∑_{j=0}^{∞} a_{j} z^{j} is uniquely determined by the properties in (a) and (b).

To prove (a), let w be a complex number such that ∑_{j=0}^{∞} a_{j} w^{j} converges. If z ∈ C satisfies |z| < |w|, then ∑_{j=0}^{∞} a_{j} z^{j} converges absolutely, by Proposition 3.8.3. This implies that |w| is a lower bound for E_{1}, so that |w| ≤ inf E_{1} = ρ when E ≠ ∅. If E = ∅, then |w| < ρ = +∞ holds trivially.

If ∑_{j=0}^{∞} a_{j} w^{j} does not converge absolutely, then ∑_{j=0}^{∞} a_{j} z^{j} does not converge for any z ∈ C with |w| < |z|, by Proposition 3.8.3 again. This implies that E ≠ ∅, and that ρ = inf E_{1} ≤ |w|.

If 0 ≤ ρ ≤ +∞ satisfies the conditions in (a) and (b), then E = ∅ when ρ = +∞, and E ≠ ∅, inf E_{1} = ρ when ρ < ∞. Alternatively, if 0 ≤ ρ′, ρ′′ ≤ +∞ both satisfy both (a) and (b) in place of ρ, then one can check directly that ρ′ = ρ′′.

3.9 Cauchy’s condensation test

Theorem 3.9.1 (Cauchy’s condensation test) Let {a_{j}}_{j=1}^{∞} be a monotonically decreasing sequence of nonnegative real numbers. Under these conditions, ∑_{j=1}^{∞} a_{j} converges if and only if ∑_{k=0}^{∞} 2^{k} a_{2^{k}} converges.

Suppose first that ∑_{k=0}^{∞} 2^{k} a_{2^{k}} converges, so that the corresponding sequence of partial sums ∑_{k=0}^{n} 2^{k} a_{2^{k}} are bounded. We would like to show that the partial sums for ∑_{j=1}^{∞} a_{j} are bounded, which would imply that the series converges.

If k is any nonnegative integer, then

∑_{j=2^{k}}^{2^{k+1}−1} a_{j} ≤ ∑_{j=2^{k}}^{2^{k+1}−1} a_{2^{k}} = 2^{k} a_{2^{k}}. (3.9.2)

If l is a positive integer, n is a nonnegative integer, and l < 2^{n+1}, then we get that

∑_{j=1}^{l} a_{j} ≤ ∑_{k=0}^{n} ∑_{j=2^{k}}^{2^{k+1}−1} a_{j} ≤ ∑_{k=0}^{n} 2^{k} a_{2^{k}}. (3.9.3)

Thus the boundedness of the partial sums of ∑_{k=0}^{∞} 2^{k} a_{2^{k}} implies the boundedness of the partial sums for ∑_{j=1}^{∞} a_{j}, as desired.

Similarly, if k is a positive integer, then

2^{k−1} a_{2^{k}} ≤ ∑_{j=2^{k−1}+1}^{2^{k}} a_{j}. (3.9.4)

If n is a positive integer, then we obtain that

∑_{k=0}^{n} 2^{k} a_{2^{k}} = a_{1} + ∑_{k=1}^{n} 2^{k} a_{2^{k}} ≤ a_{1} + 2 ∑_{k=1}^{n} ∑_{j=2^{k−1}+1}^{2^{k}} a_{j} ≤ 2 ∑_{j=1}^{2^{n}} a_{j}. (3.9.5)

It follows that the partial sums of ∑_{k=0}^{∞} 2^{k} a_{2^{k}} are bounded when the partial sums of ∑_{j=1}^{∞} a_{j} are bounded, as desired.

Let p be a positive real number, and note that a_{j} = 1/j^{p} defines a monotonically descreasing sequence of positive real numbers. In this situation,

2^{k} a_{2^{k}} = 2^{(1−p) k} = (2^{1−p})^{k}. (3.9.6)

Thus the Cauchy condensation test implies that ∑_{j=1}^{∞} 1/j^{p} converges exactly when

∑_{k=0}^{∞}(2^{1−p})^{k} (3.9.7)

converges. The latter is a geometric series that converges exactly when

2^{1−p} < 1, (3.9.8)

which holds if and only if p > 1.

3.10 Another criterion for convergence

Theorem 3.10.1 Let {a_{j}}_{j=0}^{∞} be a sequence of complex numbers, and suppose that the sequence of sums

A_{n} = ∑_{j=0}^{n} a_{j} (3.10.2)

is bounded. Also let {b_{j}}_{j=0}^{∞} be a monotonically decreasing sequence of nonnegative real numbers that converges to 0. Under these conditions, ∑_{j=0}^{∞} a_{j} b_{j} converges.

Put A_{−1} = 0, so that a_{j} = A_{j} − A_{j−1} for every nonnegative integer j. Let n be a nonnegative integer, and observe that

∑_{j=0}^{n} a_{j} b_{j} = ∑_{j=0}^{n}(A_{j} − A_{j−1}) b_{j} = ∑_{j=0}^{n} A_{j} b_{j} − ∑_{j=0}^{n} A_{j−1} b_{j}. (3.10.3)

The last sum on the right can be reexpressed as

∑_{j=0}^{n} A_{j−1} b_{j} = ∑_{j=−1}^{n−1} A_{j} b_{j+1} = ∑_{j=0}^{n} A_{j} b_{j+1} − A_{n} b_{n+1}. (3.10.4)

Thus

(3.10.5)  ∑_{j=0}^{n} a_{j} b_{j} = ∑_{j=0}^{n} A_{j} b_{j} − ∑_{j=0}^{n} A_{j} b_{j+1} + A_{n} b_{n+1}

= ∑_{j=0}^{n} A_{j} (b_{j} − b_{j+1}) + A_{n} b_{n+1}.

Lemma 3.3.6 implies that {A_{n} b_{n+1}}_{n=0}^{∞} converges to 0, because {A_{n}}_{n=0}^{∞} is a bounded sequence, and {b_{n+1}}_{n=0}^{∞} converges to 0.

It suffices to show that

∑_{j=0}^{∞} A_{j} (b_{j} − b_{j+1}) (3.10.6)

converges, by the remarks in the preceding paragraph. If n is a nonnegative integer, then

∑_{j=0}^{n}(b_{j} − b_{j+1}) = ∑_{j=0}^{n} b_{j} − ∑_{j=0}^{n} b_{j+1} = ∑_{j=0}^{n} b_{j} − ∑_{j=1}^{n+1} b_{j} = b_{0} − b_{n+1}. (3.10.7)

This implies that ∑_{j=0}^{∞}(b_{j} − b_{j+1}) converges, because {b_{n+1}}_{n=0}^{∞} converges as a sequence of real numbers. Because {A_{j}}_{j=0}^{∞} is bounded, there is a nonnegative real number C such that |A_{j}| ≤ C (3.10.8) for every j ≥ 0. It follows that

|A_{j} (b_{j} − b_{j+1})| ≤ C (b_{j} − b_{j+1}) (3.10.9)

for every j ≥ 0, because b_{j} − b_{j+1} ≥ 0, by hypothesis. Of course,

∑_{j=0}^{∞} C (b_{j} − b_{j+1}) (3.10.10)

converges, because ∑_{j=0}^{∞}(b_{j} −b_{j+1}) converges. Thus the comparison test implies that (3.10.6) converges absolutely, as desired.

If a_{j} = (−1)^{j} for every nonnegative integer j, then A_{n} is equal to 1 when j is even, and to 0 when j is odd. In particular, {A_{n}}_{n=0}^{∞} is bounded. In this case, Theorem 3.10.1 corresponds to Leibniz’ alternating series test.

Let z be a complex number, and suppose that

a_{j} = z^{j} (3.10.11)

for every nonnegative integer j. If |z| < 1, then the comparison test implies that ∑_{j=0}^{∞} b_{j} z^{j} converges absolutely when {b_{j}}_{j=0}^{∞} is bounded. Suppose that z ≠ 1, so that

A_{n} = ∑_{j=0}^{n} z^{j} = (1 − z^{n+1})/(1 − z) (3.10.12)

for every nonnegative integer n, as in (3.7.19). If |z| = 1, then

|A_{n}| = |1 − z^{n+1}|/|1 − z| ≤ (1 + |z|^{n+1})/|1 − z| = 2/|1 − z| (3.10.13)

for every n ≥ 0, so that {A_{n}}_{n=0}^{∞} is a bounded sequence of complex numbers.

3.11 Extended real numbers

The set of extended real numbers is defined to be the set of real numbers together with two additional elements, denoted +∞ and −∞. The standard ordering on the real line can be extended to the set of extended real numbers, by putting

−∞ < x < +∞ (3.11.1)

for every x ∈ R. If A is any set of extended real numbers, then the notions of upper and lower bounds for A in the set of extended real numbers can be defined in the usual way. In particular, +∞ is automatically an upper bound for A, and −∞ is automatically a lower bound for A.

The notions of supremum or least upper bound and infimum or greatest lower bound for A in the set of extended real numbers can be defined in the same way as before too. The supremum and infimum of A in the set of extended real numbers always exists, as follows. If +∞ ∈ A, or if A ∩ R has no upper bound in R, then sup A = +∞. If +∞ ∉ A, and A ∩ R is nonempty and has an upper bound in R, then sup A is the same as the supremum of A ∩ R in R. Otherwise, if A ⊆ {−∞}, then sup A = −∞. Similarly, if −∞ ∈ A or A ∩ R has no lower bound in R, then inf A = −∞. If −∞ ∉ A and A ∩ R is nonempty and has a lower bound in R, then inf A is the same as the infimum of A ∩ R in R. If A ⊆ {+∞}, then inf A = +∞. Note that

inf A ≤ sup A (3.11.2)

when A ≠ ∅.

Sums and products of extended real numbers are defined in some situations, as follows. If x ∈ R, then we put

x + (+∞) = (+∞) + x = +∞, x + (−∞) = (−∞) + x = −∞. (3.11.3)

We also put

(+∞) + (+∞) = +∞, (−∞) + (−∞) = −∞. (3.11.4)

The sum of +∞ and −∞ is not defined. If x is a nonzero extended real number, then we put

x (+∞) = (+∞) x = +∞, x (−∞) = (−∞) x = −∞ (3.11.5)

when x > 0, and

x (+∞) = (+∞) x = −∞, x (−∞) = (−∞) x = +∞ (3.11.6)

when x < 0. The product of 0 with ±∞ is not defined. We put x/(+∞) = x/(−∞) = 0 when x ∈ R, and these quotients are not defined when x = ±∞. Although 1/0 is not defined, it may be appropriate to interpret it as being +∞, when dealing with nonnegative extended real numbers.

Let {x_{j}}_{j=1}^{∞} be a sequence of real numbers.

Definition 3.11.7 We say that x_{j} tends to +∞ as j → ∞, or x_{j} → +∞ as j → ∞, if for every nonnegative real number R there is a positive integer L such that (3.11.8) x_{j} > R for every j ≥ L.

Similarly, we say that x_{j} tends to −∞ as j → ∞, or x_{j} → −∞ as j → ∞, if for every nonnegative real number R there is a positive integer L such that

(3.11.9) x_{j} < −R for every j ≥ L.

Thus, if x is any extended real number, then the condition that x_{j} → x as j → ∞ is defined, using convergence with respect to the standard Euclidean metric on R when x ∈ R, and the previous definition when x = ±∞. It is easy to see that this can hold for at most one x.

Proposition 3.11.10 Let {x_{j}}_{j=1}^{∞}, {y_{j}}_{j=1}^{∞} be sequences of real numbers, let x, y be extended real numbers, and suppose that x_{j} → x and y_{j} → y as j → ∞.

(a) If x + y is defined as an extended real number, then x_{j} + y_{j} → x + y as j → ∞.

(b) If x y is defined as an extended real number, then x_{j} y_{j} → x y and j → ∞.

Of course, this follows from previous results when x, y ∈ R. It is not difficult to prove (a) and (b) directly in the other cases. Suppose that x = +∞, for instance. If the set of y_{j}’s has a lower bound in R, then it is easy to see that x_{j} + y_{j} → ∞ as j → ∞. In particular, this holds when y_{j} → y as j → ∞ and y ≠ −∞. Similarly, suppose that there is a positive real number a and a positive integer L_{0} such that y_{j} ≥ a (3.11.11) for every j ≥ L_{0}. Using this, one can check that x_{j} y_{j} → +∞ as j → ∞. If y_{j} → y as j → ∞, then this condition holds when y > 0.

If {x_{j}}_{j=1}^{∞} is a sequence of nonzero real numbers such that |x_{j}| → ∞ as j → ∞, then 1/x_{j} → 0 as j → ∞. (3.11.12) If {x_{j}}_{j=1}^{∞} is a sequence of positive real numbers that converges to 0, then

1/x_{j} → +∞ as j → ∞. (3.11.13)

Proposition 3.11.14 If {x_{j}}_{j=1}^{∞} is any sequence of real numbers, then there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} and an extended real number x such that x_{j_{l}} → x as l → ∞.

Suppose first that there are real numbers a and b such that

(3.11.15) a ≤ x_{j} ≤ b for every j ≥ 1.

Remember that [a, b] is a compact subset of the real line with respect to the standard Euclidean metric, which implies that [a, b] has the limit point property, and hence is sequentially compact, by previous results. This means that there is a subsequence of {x_{j}}_{j=1}^{∞} that converges to an element of [a, b], as desired.

Next, suppose that the set of x_{j}’s has no upper bound in R. This implies that for each positive integer n,

(3.11.16) x_{j} > n for infinitely many j ≥ 1,

since otherwise the set of x_{j}’s would have a finite upper bound. We would like to find a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} such that

(3.11.17) x_{j_{l}} > l for every l ≥ 1,

which implies that x_{j_{l}} → ∞ as j → ∞. We can start by choosing j_{1} to be any positive integer such that x_{j_{1}} > 1. If j_{l} has been chosen for some positive integer l, then can choose j_{l+1} to be a positive integer such that j_{l+1} > j_{l} and x_{j_{l+1}} > l + 1, using (3.11.16).

Similarly, if the set of x_{j}’s does not have a lower bound in R, then one can find a subsequence that tends to −∞. This can be shown in essentially the same way, or by reducing to the previous case.

3.12 Limits superior and inferior

Let {x_{j}}_{j=1}^{∞} be a sequence of real numbers, and let E be the set of extended real numbers x for which there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} such that

x_{j_{l}} → x as l → ∞. (3.12.1)

Note that E ≠ ∅, by Proposition 3.11.14.

Definition 3.12.2 The upper limit or limit superior of {x_{j}}_{j=1}^{∞} is defined by

(3.12.3)  lim sup_{j→∞} x_{j} = sup E.

Similarly, the lower limit or limit inferior of {x_{j}}_{j=1}^{∞} is defined by

(3.12.4)  lim inf_{j→∞} x_{j} = inf E.

Thus

(3.12.5)  lim inf_{j→∞} x_{j} ≤ lim sup_{j→∞} x_{j}.

If there is an extended real number x such that x_{j} → x as j → ∞, then

E = {x}, (3.12.6)

and

(3.12.7)  lim sup_{j→∞} x_{j} = lim inf_{j→∞} x_{j} = x.

Proposition 3.12.8 Put y = lim sup_{j→∞} x_{j}.

(a) If z ∈ R satisfies y < z, then x_{j} < z for all but finitely many j ≥ 1.

(b) If w ∈ R satisfies w < y, then x_{j} > w for infinitely many j ≥ 1.

(c) There is only one extended real number y that satisfies the conditions in (a) and (b).

Suppose for the sake of a contradiction that z ∈ R, y < z, and x_{j} ≥ z for infinitely many j ≥ 1. Equivalently, this means that there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} such that

(3.12.9) x_{j_{l}} ≥ z for every l ≥ 1.

Using Proposition 3.11.14, we can get a subsequence {x_{j_{ln}} }_{n=1}^{∞} of {x_{j_{l}}}_{l=1}^{∞} and an extended real number x such that x_{j_{ln}} → x as n → ∞. It is easy to see that {x_{j_{ln}} }_{n=1}^{∞} may also be considered as a subsequence of {x_{j}}_{j=1}^{∞}, so that x ∈ E. One can also check that x ≥ z, because x_{j_{ln}} ≥ z for every n ≥ 1, by (3.12.9). Thus x > y, because y < z, by hypothesis. This contradicts the fact that y = sup E, as desired.

If w < y, then w is not an upper bound for E, by definition of the supremum. This means that there is an x ∈ E such that w < x. In this case, there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} such that x_{j_{l}} → x as l → ∞. It is easy to see that x_{j_{l}} > w for all but finitely many l ≥ 1, because w < x. In particular, x_{j} > w for infinitely many j ≥ 1, as desired.

Suppose that for the sake of a contradiction that y′ and y′′ are distinct extended real numbers that satisfy the conditions in (a) and (b) in place of y. We may as well suppose that y′ < y′′, since otherwise we can interchange the roles of y′ and y′′. This implies that there is a real number r such that y′ < r < y′′. Using (a) for y′ and z = r, we get that x_{j} < r for all but finitely many j ≥ 1. However, part (b) for y′′ and w = r implies that x_{j} > r for infinitely many j ≥ 1, which is a contradiction.

Proposition 3.12.10 Put u = lim inf_{j→∞} x_{j}.

(a) If t ∈ R satisfies t < u, then x_{j} > t for all but finitely many j ≥ 1.

(b) If v ∈ R satisfies u < v, then x_{j} < v for infinitely many j ≥ 1.

(c) There is only one extended real number u that satisfies the conditions in (a) and (b).

This can be shown in essentially the same way as before. Alternatively, one can check that (3.12.11) − lim inf_{j→∞} x_{j} = lim sup_{j→∞}(−x_{j}),

and use this to reduce to the previous proposition.

Corollary 3.12.12 Suppose that

(3.12.13)  lim sup_{j→∞} x_{j} = lim inf_{j→∞} x_{j},

and let x be their common value. Under these conditions, x_{j} → x as j → ∞.

This can be verified using part (a) of each of Propositions 3.12.8 and 3.12.10. The definitions of the upper and lower limits used here correspond to Defini-

tion 3.16 on p56 of [112]. The characterization of the upper limit in Proposition 3.12.8 is analogous to Theorem 3.17 on p56 of [112]. However, instead of part (b) of Proposition 3.12.8, part (a) of Theorem 3.17 in [112] states that

y = lim sup_{j→∞} x_{j} ∈ E. (3.12.14)

It is easy to see that this implies part (b) of Proposition 3.12.8. Conversely, it is not too difficult to show that (3.12.14) can be obtained from parts (a) and (b) of Proposition 3.12.8.

Although part (c) of Proposition 3.12.8 is stated in the same way as the last part of Theorem 3.17 in [112], its meaning is a bit different, because it uses part (b) of Proposition 3.12.8 instead of (3.12.14).

Sometimes upper and lower limits are defined or characterized in other ways, as in Definition 1.13 on p14 of [111]. In that definition, it is mentioned that the upper limit satisfies (3.12.14) and is the largest extended real number with this property, which means that the definition in [111] is equivalent to the one used in [112]. This could also be obtained from Proposition 3.12.8.

If {a_{j}}_{j=1}^{∞} and {b_{j}}_{j=1}^{∞} are sequences of real numbers, then it is well known that

(3.12.15)  lim sup_{j→∞}(a_{j} + b_{j}) ≤ lim sup_{j→∞} a_{j} + lim sup_{j→∞} b_{j},

as long as the right side is defined as an extended real number, which is to say that it is not a sum of +∞ and −∞. This corresponds to part (b) of Exercise 4 on p32 of [111], and to Exercise 5 on p78 of [112].

3.13 Applications for infinite series

Proposition 3.13.1 (The root test) Let

∑_{j=1}^{∞} a_{j} (3.13.2)

be an infinite series of complex numbers, and put

α = lim sup_{j→∞} |a_{j}|^{1/j}. (3.13.3)

The series (3.13.2) converges absolutely when α < 1, and does not converge when α > 1.

Suppose first that α < 1, and let β be a real number such that α < β < 1. Using part (a) of Proposition 3.12.8, we get that

|a_{j}|^{1/j} < β (3.13.4)

for all but finitely many j ≥ 1. Equivalently, this means that

|a_{j}| < β^{j} (3.13.5)

for all but finitely many j ≥ 1. It is easy to see that ∑_{j=1}^{∞} |a_{j}| converges in this case, in comparison with the convergent geometric series ∑_{j=1}^{∞} β^{j}.

Now suppose that α > 1, and let γ be a real number such that 1 ≤ γ < α. Part (b) of Proposition 3.12.8 implies that

|a_{j}|^{1/j} > γ (3.13.6)

for infinitely many j ≥ 1. This is the same as saying that

|a_{j}| > γ^{j} (3.13.7)

for infinitely many j ≥ 1. If we take γ = 1, then we get that {a_{j}}_{j=1}^{∞} does not converge to 0 as a sequence of complex numbers, so that ∑_{j=1}^{∞} a_{j} does not converge. We can also take γ > 1, to obtain that {a_{j}}_{j=1}^{∞} is not even bounded.

If a_{j} = 1 for every j ≥ 1, then α = 1, and ∑_{j=1}^{∞} a_{j} does not converge. If

a_{j} = 1/j^{2} for every j ≥ 1, then ∑_{j=1}^{∞} a_{j} converges, and a_{j}^{1/j} = (j^{1/j})^{−2} → 1 as j → ∞, so that α = 1.

As an application of the root test, let

∑_{j=0}^{∞} a_{j} z^{j} (3.13.8)

be a power series with complex coefficients, which obviously converges absolutely when z = 0. Also let α be as in (3.13.3) again, which can be defined without using the j = 0 term. If z is a nonzero complex number, then one can check that

lim sup_{j→∞} |a_{j} z^{j}|^{1/j} = |z| α. (3.13.9)

The root test implies that (3.13.8) converges absolutely when |z|α < 1, and that (3.13.8) does not converge when |z|α > 1. It follows that

ρ = 1/α (3.13.10)

is the radius of convergence of (3.13.8), which is interpreted as being +∞ when α = 0.

Proposition 3.13.11 (The ratio test) An infinite series ∑_{j=1}^{∞} a_{j} of nonzero complex numbers converges absolutely when

lim sup_{j→∞}(|a_{j+1}|/|a_{j}|) < 1, (3.13.12)

and does not converge when

lim inf_{j→∞}(|a_{j+1}|/|a_{j}|) > 1. (3.13.13)

If (3.13.12) holds, then let β be a real number such that

lim sup_{j→∞}(|a_{j+1}|/|a_{j}|) < β < 1. (3.13.14)

Part (a) of Proposition 3.12.8 implies that there is a positive integer L_{1} such that |a_{j+1}|/|a_{j}| < β (3.13.15)

for every j ≥ L_{1}. This means that

|a_{j+1}| < β |a_{j}| (3.13.16)

for every j ≥ L_{1}, so that |a_{j}| ≤ β^{j−L_{1}} |a_{L_{1}}| (3.13.17)

for every j ≥ L_{1}. Using this, the convergence of ∑_{j=1}^{∞} |a_{j}| can be obtained from the convergence of the geometric series ∑_{j=1}^{∞} β^{j}.

3.14. REARRANGEMENTS OF INFINITE SERIES 57

If (3.13.13) holds, then let γ be a real number such that

lim inf_{j→∞}(|a_{j+1}|/|a_{j}|) > γ ≥ 1. (3.13.18)

Part (a) of Proposition 3.12.10 implies that there is a positive integer L_{2} such that |a_{j+1}|/|a_{j}| > γ (3.13.19) for every j ≥ L_{2}. Thus |a_{j+1}| > γ |a_{j}| (3.13.20) for every j ≥ L_{2}, and hence |a_{j}| ≥ γ^{j−L_{2}} |a_{L_{2}}| (3.13.21) for every j ≥ L_{2}. In particular, this implies that {a_{j}}_{j=1}^{∞} does not converge to 0 when γ = 1, so that ∑_{j=1}^{∞} a_{j} does not converge. If we take γ > 1, then we get that |a_{j}| → ∞ as j → ∞.

3.14 Rearrangements of infinite series

Let π be a one-to-one mapping from the set of positive integers onto itself. Thus π(j) ∈ Z_{+} for every j ∈ Z_{+}, and every positive integer k can be expressed as π(j) for exactly one j ∈ Z_{+}, which may be denoted π^{−1}(k). If ∑_{j=1}^{∞} a_{j} is an infinite series of complex numbers, then ∑_{j=1}^{∞} a_{π(j)} is called a rearrangement of ∑_{j=1}^{∞} a_{j}. If a_{j} = 0 for all but finitely many j ≥ 1, then it is easy to see that a_{π(j)} = 0 for all but finitely many j ≥ 1 too, and that

∑_{j=1}^{∞} a_{π(j)} = ∑_{j=1}^{∞} a_{j}. (3.14.1)

Suppose for the moment that a_{j} is a nonnegative real number for each j ≥ 1. If l, n are positive integers such that max{π(j) : 1 ≤ j ≤ l} ≤ n, (3.14.2) then

∑_{j=1}^{l} a_{π(j)} ≤ ∑_{j=1}^{n} a_{j}. (3.14.3)

Similarly, if max{π^{−1}(j) : 1 ≤ j ≤ l} ≤ n, (3.14.4) then

∑_{j=1}^{l} a_{j} ≤ ∑_{j=1}^{n} a_{π(j)}. (3.14.5)

If ∑_{j=1}^{∞} a_{j} converges, then one can use (3.14.3) and (3.14.5) to check that ∑_{j=1}^{∞} a_{π(j)} converges, and that (3.14.1) holds.

If ∑_{j=1}^{∞} a_{j} is an absolutely convergent series of complex numbers, then it is easy to see that ∑_{j=1}^{∞} a_{π(j)} is absolutely convergent as well, by applying the remarks in the preceding paragraph to ∑_{j=1}^{∞} |a_{j}|. In order to show that (3.14.1) holds in this situation, suppose first that a_{j} is a real number for each j ≥ 1. In this case, one can express ∑_{j=1}^{∞} a_{j} as a difference of convergent series of nonnegative real numbers, to which the previous remarks can be applied. If the a_{j}’s are complex numbers, then one can reduce to the real case, by considering the real and imaginary parts.

3.15 Cauchy products of infinite series

Let ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} be infinite series of complex numbers. Put

c_{n} = ∑_{j=0}^{n} a_{j} b_{n−j} (3.15.1)

for every nonnegative integer n. The corresponding infinite series ∑_{n=0}^{∞} c_{n} is called the Cauchy product of ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l}. If a_{j} = 0 for all but finitely many j ≥ 0, and b_{l} = 0 for all but finitely many l ≥ 0, then one can verify that c_{n} = 0 for all but finitely many n ≥ 0, and that

∑_{n=0}^{∞} c_{n} = ( ∑_{j=0}^{∞} a_{j}) ( ∑_{l=0}^{∞} b_{l}). (3.15.2)

Suppose for the moment that a_{j} is a nonnegative real number for every j ≥ 0, and that b_{l} is a nonnegative real number for every l ≥ 0, so that c_{n} is a nonnegative real number for every n ≥ 0. If N is a nonnegative integer, then it is easy to see that

∑_{n=0}^{N} c_{n} ≤ ( ∑_{j=0}^{N} a_{j}) ( ∑_{l=0}^{N} b_{l}). (3.15.3)

Similarly, if N_{1} and N_{2} are nonnegative integers, then one can check that

( ∑_{j=0}^{N_{1}} a_{j}) ( ∑_{l=0}^{N_{2}} b_{l}) ≤ ∑_{n=0}^{N_{1}+N_{2}} c_{n}. (3.15.4)

If ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge, then one can use (3.15.3) to get that ∑_{n=0}^{∞} c_{n} converges, with

∑_{n=0}^{∞} c_{n} ≤ ( ∑_{j=0}^{∞} a_{j}) ( ∑_{l=0}^{∞} b_{l}). (3.15.5)

In this case, one can also use (3.15.4) to get that

( ∑_{j=0}^{∞} a_{j}) ( ∑_{l=0}^{∞} b_{l}) ≤ ∑_{n=0}^{∞} c_{n}, (3.15.6)

so that (3.15.2) holds.

If the a_{j}’s and b_{l}’s are complex numbers, then

|c_{n}| ≤ ∑_{j=0}^{n} |a_{j}| |b_{n−j}| (3.15.7)

for every n ≥ 0. The sum on the right is the same as the nth term of the Cauchy product of ∑_{j=0}^{∞} |a_{j}| and ∑_{l=0}^{∞} |b_{l}|. If these two series converge, then their Cauchy product converges too, as in the preceding paragraph. Under these conditions, ∑_{n=0}^{∞} |c_{n}| converges as well, by the comparison test. More precisely, we have that

∑_{n=0}^{∞} |c_{n}| ≤ ( ∑_{j=0}^{∞} |a_{j}|) ( ∑_{l=0}^{∞} |b_{l}|). (3.15.8)

Thus ∑_{n=0}^{∞} c_{n} converges absolutely when ∑_{j=0}^{∞} a_{j} and ∑_{l=0}^{∞} b_{l} converge absolutely. One can check that (3.15.2) holds in this situation, by reducing to the case of convergent series of nonnegative real numbers, as in the previous section.

Continuous mappings

4.1 Continuity at a point

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces, and let f be a function defined on M with values in N. This is also known as a mapping from M into N, which may be expressed by f : M → N.

Definition 4.1.1 We say that f is continuous at a point x ∈ M if for every ϵ > 0 there is a δ > 0 such that

ρ(f(x), f(y)) < ϵ (4.1.2)

for every y ∈ M with d(x, y) < δ.

Proposition 4.1.3 A mapping f from M into N is continuous at x ∈ M if and only if for every sequence {x_{j}}_{j=1}^{∞} of elements of M that converges to x, {f(x_{j})}_{j=1}^{∞} converges to f(x) in N.

Suppose first that f is continuous at x, and let {x_{j}}_{j=1}^{∞} be a sequence of elements of M that converges to x. Also let ϵ > 0 be given, and let δ > 0 be as in Definition 4.1.1. Because {x_{j}}_{j=1}^{∞} converges to x in M, there is a positive integer L such that d(x, x_{j}) < δ (4.1.4)

for every j ≥ L. This implies that

ρ(f(x), f(x_{j})) < ϵ (4.1.5)

for every j ≥ L, by (4.1.2), as desired.

Conversely, we would like to show that f is continuous at x when it has this property in terms of sequences. Let ϵ > 0 be given, and suppose for the sake of a contradiction that there is no δ > 0 as in Definition 4.1.1. This means that for every δ > 0 there is a point x(δ) ∈ M such that d(x, x(δ)) < δ and

ρ(f(x), f(x(δ))) ≥ ϵ. (4.1.6)

If j is a positive integer, then we can apply this with δ = 1/j, to get a point x_{j} ∈ M such that d(x, x_{j}) < 1/j (4.1.7)

and ρ(f(x), f(x_{j})) ≥ ϵ. (4.1.8)

This leads to a sequence {x_{j}}_{j=1}^{∞} of elements of M that converges to x in M, and for which {f(x_{j})}_{j=1}^{∞} does not converge to f(x) in N, as desired.

Proposition 4.1.9 Let (M, d(·, ·)) be a metric space, and let f, g be complexvalued functions on M. Suppose that f and g are continuous at a point x ∈ M, with respect to the standard metric on C. Under these conditions, f + g and f g are continuous at x as well. If, for every w ∈ M, f(w) ≠ 0, then 1/f is continuous at x too.

This follows from Proposition 4.1.3, and previous results about convergent sequences of complex numbers.

Definition 4.1.10 Let (M, d(·, ·)) and (N, ρ(·, ·)) be metric spaces. A mapping f from M into N is said to be continuous on M if f is continuous at every point x ∈ M.

One can use Proposition 4.1.9 to show that

(4.1.11) polynomial functions on R^{n} are continuous,

with respect to the standard Euclidean metric on R^{n}. Similarly,

(4.1.12) rational functions are continuous,

when the denominator is not zero.

If z and w are complex numbers, then one can check that

||z| − |w|| ≤ |z − w|, (4.1.13)

using the triangle inequality. One can use this to show that the absolute value defines a continuous real-valued function on the complex plane, with respect to the standard Euclidean metrics on R and C.

4.2 Compositions and inverse images

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces.

Proposition 4.2.1 A mapping f from M into N is continuous if and only if for every open set V ⊆ N, its inverse image

f^{−1}(V) = {x ∈ M : f(x) ∈ V } (4.2.2)

is an open set in M.

Suppose that f is continuous, and let an open set V ⊆ N be given. Let x ∈ f^{−1}(V) be given, so that f(x) ∈ V. Because V is an open set in N, there is an ϵ > 0 such that

{z ∈ N : ρ(f(x), z) < ϵ} ⊆ V. (4.2.3)

Using the continuity of f at x, we get that there is a δ > 0 such that if y ∈ M satisfies d(x, y) < δ, then ρ(f(x), f(y)) < ϵ, and hence f(y) ∈ V. This shows that {y ∈ M : d(x, y) < δ} ⊆ f^{−1}(V), (4.2.4)

as desired.

Conversely, suppose that f has the property described in the proposition, and let us show that f is continuous on M. Let x ∈ M be given, and let us show that f is continuous at x. To do this, let ϵ > 0 be given, and note that

V = {z ∈ N : ρ(f(x), z) < ϵ} (4.2.5)

is an open set in N, because it is an open ball. Thus f^{−1}(V) is an open set in M, by hypothesis. Of course, f(x) ∈ V, by construction, so that x ∈ f^{−1}(V). It follows that there is a δ > 0 such that (4.2.4) holds, by the definition of an open set. If y ∈ M satisfies d(x, y) < δ, then we get that y ∈ f^{−1}(V), so that f(y) ∈ V, as desired.

Corollary 4.2.6 A mapping f from M into N is continuous if and only if for every closed set E ⊆ N, f^{−1}(E) is a closed set in M.

It is easy to see that

f^{−1}(N \ E) = M \ f^{−1}(E) (4.2.7)

for every E ⊆ N. Using this, one can check that the condition in Corollary 4.2.6 if equivalent to the one in Proposition 4.2.1, because a subset of a metric space is a closed set if and only if its complement is an open set.

Proposition 4.2.8 Let (M_{1}, d_{1}(·, ·)), (M_{2}, d_{2}(·, ·)), and (M_{3}, d_{3}(·, ·)) be metric spaces, let f be a continuous mapping from M_{1} into M_{2}, and let g be a continuous mapping from M_{2} into M_{3}. Under these conditions, the composition g ∘ f of f and g is continuous as a mapping from M_{1} into M_{3}.

More precisely, g ∘ f is defined by putting (g ∘ f)(w) = g(f(w)) for every w ∈ M_{1}.

To prove the proposition, let x ∈ M_{1} be given, and let us check that g ∘ f is continuous at x. Let ϵ > 0 be given, and let us use the continuity of g at f(x) to get that there is an η > 0 such that

d_{3}(g(f(x)), g(u)) < ϵ (4.2.9)

4.3. IMAGES OF COMPACT SETS 63 for every u ∈ M_{2} with d_{2}(f(x), u) < η. The continuity of f at x implies that there is a δ > 0 such that d_{2}(f(x), f(y)) < η (4.2.10) for every y ∈ M_{1} with d_{1}(x, y) < δ. In this case, we can take u = f(y) in (4.2.9), to get that d_{3}(g(f(x)), g(f(y))) < ϵ, (4.2.11) as desired.

Alternatively, if {x_{j}}_{j=1}^{∞} is any sequence of elements of M_{1} that converges to x, then (4.2.12) {f(x_{j})}_{j=1}^{∞} converges to f(x) in M_{2}, because f is continuous at x. This implies that (4.2.13) {g(f(x_{j}))}_{j=1}^{∞} converges to g(f(x)) in M_{3}, because g is continuous at f(x), as desired.

We can also use the characterization of continuity in Proposition 4.2.1. Let W be an open subset of M_{3}, so that (4.2.14) g^{−1}(W) is an open subset of M_{2}, by the continuity of g. This implies that (4.2.15) f^{−1}(g^{−1}(W)) is an open set in M_{1}, by the continuity of f. It is easy to see that (g ∘ f)^{−1}(W) = f^{−1}(g^{−1}(W)), (4.2.16) directly from the definitions. Thus (4.2.17) (g ∘ f)^{−1}(W) is an open set in M_{1}, as desired.

4.3 Images of compact sets

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces. Theorem 4.3.1 If f is a continuous mapping from M into N and K is a compact subset of M, then f(K) = {f(x) : x ∈ K} (4.3.2) is a compact subset of N.

Let {U_{α}}_{α∈A} be an arbitrary open covering of f(K) in N. Thus U_{α} is an open subset of N for every α ∈ A, so that f^{−1}(U_{α}) is an open set in M for every α ∈ A, by Proposition 4.2.1. One can check that

K ⊆ ⋃_{α∈A} f^{−1}(U_{α}), (4.3.3)

because f(K) ⊆ ⋃_{α∈A} U_{α}, by hypothesis. This shows that {f^{−1}(U_{α})}_{α∈A} is an open covering of K in M. If K is compact in M, then it follows that there are finitely many indices α_{1}, …, α_{n} ∈ A such that

K ⊆ ⋃_{j=1}^{n} f^{−1}(U_{α_{j}}). (4.3.4)

Using this, one can verify that

f(K) ⊆ ⋃_{j=1}^{n} U_{α_{j}}, (4.3.5)

as desired.

Alternatively, suppose that K is sequentially compact in M, and let us show that (4.3.6) f(K) is sequentially compact in N.

Let {z_{j}}_{j=1}^{∞} be an arbitrary sequence of elements of f(K). If j is any positive integer, then let us choose x_{j} ∈ K such that

f(x_{j}) = z_{j}. (4.3.7)

Because K is sequentially compact, there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} that converges to an element x of K in M. Thus

{z_{j_{l}}}_{l=1}^{∞} = {f(x_{j_{l}})}_{l=1}^{∞} (4.3.8)

is a subsequence of {z_{j}}_{j=1}^{∞} that converges to f(x) ∈ f(K) in N, because f is continuous at x, by hypothesis.

Corollary 4.3.9 (Extreme value theorem) Suppose that f is a continuous real-valued function on M, with respect to the standard Euclidean metric on R. If K is a nonempty compact subset of M, then there are points p, q ∈ K such that f(p) ≤ f(x) ≤ f(q) (4.3.10)

for every x ∈ M.

Theorem 4.3.1 implies that f(K) is a compact subset of the real line, so that f(K) is closed and bounded. Of course, f(K) ≠ ∅, because K ≠ ∅, by hypothesis, so that the supremum and infimum of f(K) exist in R. In this situation, the supremum and infimum of f(K) are elements of f(K), because f(K) is a closed set in R, as in Proposition 2.8.8. This is the same as saying that the maximum and minimum of f are attained on K, as desired.

4.4 Uniform continuity

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces.

Definition 4.4.1 A mapping f from M into N is said to be uniformly continuous on M if for every ϵ > 0 there is a δ > 0 such that

ρ(f(x), f(y)) < ϵ (4.4.2)

for every x, y ∈ M with d(x, y) < δ.

Note that uniform continuity implies ordinary continuity. In order to get an example where the converse does not hold, one can take M = N = R with the standard Euclidean metric, and put f(x) = x^{2} for every x ∈ R. One can check that f is not uniformly continuous on R, using the fact that

f(x) − f(y) = x^{2} − y^{2} = (x + y) (x − y) (4.4.3)

for every x, y ∈ R.

Alternatively, let us take M to be the open unit interval (0, 1) in the real line, equipped with the restriction of the standard Euclidean metric on R to (0, 1). If we put f(x) = 1/x for every x ∈ (0, 1), then one can verify that f is not uniformly continuous on (0, 1).

Let (M, d(x, y)) and (N, ρ(u, v)) be arbitrary metric spaces again.

Theorem 4.4.4 If f is a continuous mapping from M into N, and if M is compact, then f is uniformly continuous on M.

Let ϵ > 0 be given. If x ∈ M, then there is a positive real number δ(x) such that ρ(f(x), f(y)) < ϵ/2 (4.4.5)

for every y ∈ M with d(x, y) < δ(x). Let B(x) be the open ball in M centered at x with radius δ(x)/2. The collection of these open balls B(x), x ∈ M, is an open covering of M, because x ∈ B(x) for every x ∈ M, and open balls in M are open sets. If M is compact, then there are finitely many elements x_{1}, …, x_{l} of M such that

M ⊆ ⋃_{j=1}^{l} B(x_{j}). (4.4.6)

Put δ = min{δ(x_{j})/2 : 1 ≤ j ≤ l}, (4.4.7)

which is a positive real number. Let w and y be elements of M such that d(w, y) < δ. Using (4.4.6), we get that there is a j ∈ {1, …, l} such that w ∈ B(x_{j}), so that d(x_{j}, w) < δ(x_{j})/2. It follows that

d(x_{j}, y) ≤ d(x_{j}, w) + d(w, y) < δ(x_{j})/2 + δ ≤ δ(x_{j}). (4.4.8)

This implies that

ρ(f(w), f(y)) ≤ ρ(f(w), f(x_{j})) + ρ(f(x_{j}), f(y)) < ϵ/2 + ϵ/2 = ϵ, (4.4.9)

as desired.

Alternatively, let ϵ > 0 be given again, and suppose for the sake of a contradiction that there is no δ > 0 that satisfies the condition in the definition of uniform continuity. This means that for every δ > 0 there are points x(δ), y(δ) ∈ M such that d(x(δ), y(δ)) < δ and

ρ(f(x(δ)), f(y(δ))) ≥ ϵ. (4.4.10)

If j is a positive integer, then we can take δ = 1/j, to get points x_{j}, y_{j} ∈ M such that d(x_{j}, y_{j}) < 1/j and

ρ(f(x_{j}), f(y_{j})) ≥ ϵ. (4.4.11)

Because M is compact, and thus sequentially compact, there is a subsequence {x_{j_{l}}}_{l=1}^{∞} of {x_{j}}_{j=1}^{∞} that converges to x ∈ M. Using the same sequence of indices {j_{l}}_{l=1}^{∞}, we get a corresponding subsequence {y_{j_{l}}}_{l=1}^{∞} of {y_{j}}_{j=1}^{∞}. It is easy to see that {y_{j_{l}}}_{l=1}^{∞} converges to x in M in this situation as well. This implies that {f(x_{j_{l}})}_{l=1}^{∞} and {f(y_{j_{l}})}_{l=1}^{∞} both converge to f(x) in N, because f is continuous at x, by hypothesis. Of course,

ρ(f(x_{j_{l}}), f(y_{j_{l}})) ≤ ρ(f(x_{j_{l}}), f(x)) + ρ(f(x), f(y_{j_{l}})) (4.4.12)

for every l ≥ 1, by the triangle inequality. The right side of (4.4.12) is as small as we want when l is sufficiently large, because {f(x_{j_{l}})}_{l=1}^{∞} and {f(y_{j_{l}})}_{l=1}^{∞} converge to f(x) in N. This contradicts (4.4.11), as desired.

4.5 Images of connected sets

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces.

Theorem 4.5.1 If f is a continuous mapping from M into N, and if E is a connected subset of M, then f(E) is connected as a subset of N.

Suppose for the sake of a contradiction that f(E) is not connected in N. This means that there are nonempty separated subsets A, B of N such that f(E) = A ∪ B. Put

A_{1} = f^{−1}(A) ∩ E, B_{1} = f^{−1}(B) ∩ E. (4.5.2)

It is easy to see that A_{1}∪B_{1} = E, by construction. We also have that A_{1}, B_{1} ≠ ∅, because A and B are nonempty subsets of f(E).

We would like to check that A_{1} and B_{1} are separated in M. Observe that A_{1} ⊆ f^{−1}(A) ⊆ f^{−1}(\overline{A}), where \overline{A} is the closure of A in N. Remember that \overline{A} is a closed set in N, so that f^{−1}(\overline{A}) is a closed set in M, as in Corollary 4.2.6.

Using this, one can check that the closure \overline{A_{1}} of A_{1} in M is contained in f^{−1}(\overline{A}). This implies that \overline{A_{1}} ∩ B_{1} ⊆ f^{−1}(\overline{A}) ∩ f^{−1}(B). (4.5.3)

However, f^{−1}(\overline{A}) ∩ f^{−1}(B) = f^{−1}(\overline{A} ∩ B) = f^{−1}(∅) = ∅, because A and B are separated in N. It follows that \overline{A_{1}} ∩ B_{1} = ∅. One can show that A_{1} ∩ \overline{B_{1}} = ∅ in the same way, so that A_{1} and B_{1} are separated in M. This means that E is not connected in M, as desired.

Corollary 4.5.4 (Intermediate value theorem) Let a and b be real numbers with a < b, and let f be a continuous real-valued function on the closed interval [a, b], with respect to the standard Euclidean metric on R and its restriction to [a, b]. If t ∈ R satisfies f(a) < t < f(b) or f(b) < t < f(a), then there is an x ∈ (a, b) such that f(x) = t.

Let us first extend f to a real-valued function on the real line, by putting f(y) = f(a) when y ∈ R satisfies y < a, and f(z) = f(b) when z ∈ R satisfies z > b. It is easy to see that this extension is continuous on R. Remember that [a, b] is connected as a subset of the real line, as in Theorem 2.8.4. It follows that f([a, b]) is connected in R as well, by the previous theorem. If t is as in the statement of the corollary, then we get that t ∈ f([a, b]), using Theorem 2.8.4 again.

4.6 Uniform convergence

Let M be a set, and let (N, ρ(u, v)) be a metric space. Also let {f_{j}}_{j=1}^{∞} be a sequence of mappings from M into N, and let f be another mapping from M into N.

Definition 4.6.1 We say that {f_{j}}_{j=1}^{∞} converges to f pointwise on M if for every x ∈ M, {f_{j}(x)}_{j=1}^{∞} converges to f(x) in N. We say that {f_{j}}_{j=1}^{∞} converges to f uniformly on M if for every ϵ > 0 there is a positive integer L such that

ρ(f_{j}(x), f(x)) < ϵ (4.6.2)

for every x ∈ M and j ≥ L.

It is easy to see that uniform convergence implies pointwise convergence. To get an example where the converse does not hold, let us take M to be the closed unit interval [0, 1], N = R with the standard Euclidean metric, and f_{j}(x) = x^{j} for every x ∈ [0, 1] and j ≥ 1. We have seen that {f_{j}}_{j=1}^{∞} converges pointwise to the real-valued function f defined on [0, 1] by putting f(x) = 0 when 0 ≤ x < 1, and f(1) = 1. However, one can check that {f_{j}}_{j=1}^{∞} does not converge to f uniformly on [0, 1].

Theorem 4.6.3 Let (M, d(·, ·)) and (N, ρ(·, ·)) be metric spaces, and let

{f_{j}}_{j=1}^{∞} (4.6.4) be a sequence of mappings from M into N that converges uniformly to a mapping f from M into N. If x ∈ M and f_{j} is continuous at x for every j ≥ 1, then f is continuous at x too.

Let ϵ > 0 be given. Because {f_{j}}_{j=1}^{∞} converges to f uniformly on M, there is a positive integer L such that

ρ(f_{j}(w), f(w)) < ϵ/3 (4.6.5)

for every w ∈ M and j ≥ L. By hypothesis, f_{L} is continuous at x, and so there is a δ_{L} > 0 such that

ρ(f_{L}(x), f_{L}(y)) < ϵ/3 (4.6.6)

for every y ∈ M with d(x, y) < δ_{L}. It follows that

ρ(f(x), f(y)) ≤ ρ(f(x), f_{L}(x)) + ρ(f_{L}(x), f_{L}(y)) + ρ(f_{L}(y), f(y))
(4.6.7)  < ϵ/3 + ϵ/3 + ϵ/3 = ϵ

for every y ∈ M with d(x, y) < δ_{L}, as desired.

The following criterion for uniform convergence is due to Weierstrass.

Proposition 4.6.8 Let a_{1}, a_{2}, a_{3}, … be an infinite sequence of complex-valued functions on a set M, and let A_{1}, A_{2}, A_{3}, … be an infinite sequence of nonnegative real numbers. Suppose that

|a_{j}(x)| ≤ A_{j} (4.6.9)

for every x ∈ M and j ≥ 1, and that ∑_{j=1}^{∞} A_{j} converges. Note that ∑_{j=1}^{∞} a_{j}(x) converges absolutely for every x ∈ M, by the comparison test. Under these conditions, the sequence of partial sums ∑_{j=1}^{n} a_{j}(x) converges to ∑_{j=1}^{∞} a_{j}(x) uniformly on M as n → ∞, with respect to the standard metric on C.

If x ∈ M and n is a positive integer, then one can verify that

|∑_{j=1}^{∞} a_{j}(x) − ∑_{j=1}^{n} a_{j}(x)| = | ∑_{j=n+1}^{∞} a_{j}(x)| ≤ ∑_{j=n+1}^{∞} |a_{j}(x)| ≤ ∑_{j=n+1}^{∞} A_{j}. (4.6.10)

Of course,

∑_{j=n+1}^{∞} A_{j} = ∑_{j=1}^{∞} A_{j} − ∑_{j=1}^{n} A_{j} (4.6.11)

tends to 0 as n → ∞, because ∑_{j=1}^{∞} A_{j} converges. The proposition follows easily from these two statements.

4.7 Continuity of power series

Let ∑_{j=0}^{∞} c_{j} z^{j} be a power series with coefficients in C.

Proposition 4.7.1 Suppose that ∑_{j=0}^{∞} |c_{j}| r^{j} converges for some positive real number r. Consider the complex-valued function defined on the closed disk

{z ∈ C : |z| ≤ r} (4.7.2)

by

f(z) = ∑_{j=0}^{∞} c_{j} z^{j}, (4.7.3)

where the series converges absolutely by the comparison test. Under these conditions, f is continuous on (4.7.2), with respect to the standard metric on C and its restriction to (4.7.2).

If j is a nonnegative integer, then

|c_{j} z^{j}| = |c_{j}| |z|^{j} ≤ |c_{j}| r^{j} (4.7.4)

on (4.7.2). Thus the sequence of partial sums

∑_{j=0}^{n} c_{j}z^{j} (4.7.5)

converges to f(z) uniformly on (4.7.2), by Weierstrass’ criterion. We also have that (4.7.5) is continuous on the complex plane for every n ≥ 0, and in particular the restrictions of these functions to (4.7.2) are continuous. The continuity of f on (4.7.2) now follows from Theorem 4.6.3.

Proposition 4.7.6 Let R be a positive extended real number, and suppose that ∑_{j=0}^{∞} |c_{j}| r^{j} converges for every positive real number r with r < R. Consider the complex-valued function defined on

{z ∈ C : |z| < R} (4.7.7)

by (4.7.3), where the series converges absolutely by the comparison test. Under these conditions, f is continuous on (4.7.7), with respect to the standard metric on C, and its restriction to (4.7.7).

Let a point z_{0} ∈ C with |z_{0}| < R be given, and let us check that f is continuous at z_{0}, as a complex-valued function defined on (4.7.7). The restriction of f to (4.7.2) is continuous when 0 < r < R, as in the previous proposition. If |z_{0}| < r < R, then one can check that the continuity of f at z_{0} as a function on (4.7.2) implies the continuity of f at z_{0} as a function on (4.7.7).

4.8 The supremum metric

Let M be a set, and let (N, ρ(u, v)) be a metric space.

Definition 4.8.1 A mapping f from M into N is said to be bounded if f(M) is a bounded set in N.

Now let (M, d(x, y)) be a metric space as well. The space of all continuous mappings from M into N may be denoted C(M, N). The space of all mappings from M into N that are both bounded and continuous may be denoted C_{b}(M, N). If M is compact, then every continuous mapping from M into N is bounded, by Theorem 4.3.1.

Suppose from now on in this section that M ≠ ∅. If f and g are bounded continuous mappings from M into N, then one can check that ρ(f(x), g(x)) is bounded as a real-valued function on M. Put

θ(f, g) = sup{ρ(f(x), g(x)) : x ∈ M}, (4.8.2)

which is a nonnegative real number. If f(x) = g(x) for every x ∈ M, then ρ(f(x), g(x)) = 0 for every x ∈ M, and θ(f, g) = 0. Conversely, if θ(f, g) = 0, then ρ(f(x), g(x)) = 0 for every x ∈ M, so that f(x) = g(x) for every x ∈ M.

It is easy to see that θ(f, g) = θ(g, f), (4.8.3)

because ρ(·, ·) is symmetric. Let h be another bounded continuous mapping from M into N, and observe that

ρ(f(x), h(x)) ≤ ρ(f(x), g(x)) + ρ(g(x), h(x)) ≤ θ(f, g) + θ(g, h) (4.8.4)

for every x ∈ M. This implies that

θ(f, h) ≤ θ(f, g) + θ(g, h). (4.8.5)

Thus θ(·, ·) defines a metric on C_{b}(M, N), which is known as the supremum metric.

Proposition 4.8.6 A sequence {f_{j}}_{j=1}^{∞} of bounded continuous mappings from M into N converges to f ∈ C_{b}(M, N) with respect to the supremum metric if and only if {f_{j}}_{j=1}^{∞} converges to f uniformly on M.

Suppose first that {f_{j}}_{j=1}^{∞} converges to f with respect to the supremum metric, and let ϵ > 0 be given. By definition of convergence of a sequence in a metric space, there is a positive integer L such that

θ(f_{j}, f) < ϵ (4.8.7)

for every j ≥ L. If x ∈ M, then it follows that

ρ(f_{j}(x), f(x)) ≤ θ(f_{j}, f) < ϵ (4.8.8)

4.9. COMPLETENESS OF C_{B}(M, N) 71 for every j ≥ L. This means that {f_{j}}_{j=1}^{∞} converges to f uniformly on M, as desired.

Conversely, suppose that {f_{j}}_{j=1}^{∞} converges to f uniformly on M. Thus, for each ϵ > 0, there is a positive integer L(ϵ) such that ρ(f_{j}(x), f(x)) < ϵ (4.8.9) for every x ∈ M and j ≥ L(ϵ). It follows that θ(f_{j}, f) ≤ ϵ (4.8.10) for every j ≥ L(ϵ), by the definition of the supremum metric. This means that θ(f_{j}, f) ≤ ϵ/2 < ϵ (4.8.11) for every j ≥ L(ϵ/2), so that {f_{j}}_{j=1}^{∞} converges to f with respect to the supremum metric.

4.9 Completeness of C_{b}(M, N)

Let (M, d(x, y)) and (N, ρ(u, v)) be nonempty metric spaces. Theorem 4.9.1 If N is complete as a metric space with respect to ρ(·, ·), then C_{b}(M, N) is complete with respect to the supremum metric.

Let {f_{j}}_{j=1}^{∞} be a Cauchy sequence of elements of C_{b}(M, N), with respect to the supremum metric. This means that for every ϵ > 0 there is a positive integer L(ϵ) such that θ(f_{j}, f_{l}) < ϵ (4.9.2) for every j, l ≥ L(ϵ). If x ∈ M, then we get that ρ(f_{j}(x), f_{l}(x)) ≤ θ(f_{j}, f_{l}) < ϵ (4.9.3) for every j, l ≥ L(ϵ). This shows that {f_{j}(x)}_{j=1}^{∞} is a Cauchy sequence in N. It follows that {f_{j}(x)}_{j=1}^{∞} converges in N, because N is complete by hypothesis.

Put

f(x) = lim_{j→∞} f_{j}(x) (4.9.4)

for every x ∈ M, which defines a mapping from M into N. One can check that ρ(f(x), f_{l}(x)) ≤ ϵ (4.9.5) for every x ∈ M and l ≥ L(ϵ), using (4.9.3) and (4.9.4). More precisely, this uses the fact that ρ(f(x), f_{l}(x)) ≤ ρ(f(x), f_{j}(x)) + ρ(f_{j}(x), f_{l}(x)) (4.9.6) for all j ≥ 1, by the triangle inequality. This implies that

(4.9.7) {f_{l}}_{l=1}^{∞} converges to f uniformly on M.

It follows that (4.9.8) f is continuous on M,

by Theorem 4.6.3, and because f_{l} is continuous on M for every l ≥ 1, by hypothesis.

One can verify that (4.9.9) f is bounded on M,

using (4.9.5) with ϵ = 1 and l = L(1), and the boundedness of f_{L(1)} on M, by hypothesis. Thus f ∈ C_{b}(M, N), and {f_{l}}_{l=1}^{∞} converges to f with respect to the supremum metric, because of uniform convergence, as in the previous section.

4.10 The limit of a function

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces, let E be a subset of M, and suppose that p ∈ M is a limit point of E. Also let f be a function defined on E with values in N, and let q be an element of N.

Definition 4.10.1 We say that the limit of f(x), as x ∈ E approaches p, is equal to q, if for every ϵ > 0 there is a δ > 0 such that

ρ(f(x), q) < ϵ (4.10.2)

for every x ∈ E with d(p, x) < δ and x ≠ p.

In this case, we put

lim_{{x∈E \atop x→p}} f(x) = q, (4.10.3)

or simply

lim_{x→p} f(x) = q, (4.10.4)

when E = M, or it is clear which set E being used.

Proposition 4.10.5 Under the conditions just mentioned, (4.10.3) holds if and only if for every sequence {x_{j}}_{j=1}^{∞} of elements of E that converges to p in M and satisfies x_{j} ≠ p for each j ≥ 1, we have that {f(x_{j})}_{j=1}^{∞} converges to q in N.

This is analogous to Proposition 4.1.3, and we omit the details. One can use this to get the uniqueness of the limit, when it exists, from the corresponding statement for convergent sequences.

Proposition 4.10.6 Let f and g be complex-valued functions on E, and suppose that the limits of f(x) and g(x), as x ∈ E approaches p, exist, with respect to the standard metric on C. Under these conditions,

(4.10.7)  lim_{{x∈E \atop x→p}}(f(x) + g(x)) = lim_{{x∈E \atop x→p}} f(x) + lim_{{x∈E \atop x→p}} g(x)

and

(4.10.8)  lim_{{x∈E \atop x→p}}(f(x) g(x)) = ( lim_{{x∈E \atop x→p}} f(x)) ( lim_{{x∈E \atop x→p}} g(x)).

We also have that

lim_{{x∈E \atop x→p}} 1/f(x) = ( lim_{{x∈E \atop x→p}} f(x))^{−1} (4.10.9)

when f(x) ≠ 0 for every x ∈ E, and the limit on the right is not zero.

This follows from the previous proposition, and the corresponding statements for convergent sequences of complex numbers.

Let f be a mapping from M into a metric space N again. If p ∈ M is a limit point of M, then it is easy to see that f is continuous at p if and only if

lim_{x→p} f(x) = f(p). (4.10.10)

If p is not a limit point of M, then one can check that f is automatically continuous at p.

4.11 One-sided limits

Let a and b be real numbers with a < b, and let (N, ρ(u, v)) be a metric space. Also let f be a function defined on the open interval (a, b) in the real line with values in N. If p ∈ R satisfies a ≤ p < b, then put

E_{+}(p) = {x ∈ R : p < x < b}. (4.11.1)

The limit of f(x) as x ∈ (a, b) approaches p from the right is defined by

f(p+) = lim_{x→p+} f(x) = lim_{{x∈E_{+}(p) \atop x→p}} f(x), (4.11.2)

when the limit on the right side exists, with respect to the standard Euclidean metric on R. Similarly, if p ∈ R satisfies a < p ≤ b, then put

E_{−}(p) = {x ∈ R : a < x < p}. (4.11.3)

The limit of f(x) as x ∈ (a, b) approaches p from the left is defined by

f(p−) = lim_{x→p−} f(x) = lim_{{x∈E_{−}(p) \atop x→p}} f(x), (4.11.4)

when the limit on the right side exists. If p ∈ (a, b), then it is easy to see that

lim_{x→p} f(x) (4.11.5)

exists if and only if the one-sided limits (4.11.2) and (4.11.4) exist and are equal, in which case (4.11.5) is equal to the common value of the one-sided limits.

Definition 4.11.6 A real-valued function f on (a, b) is said to be monotonically increasing on (a, b) if for every x, y ∈ (a, b) with x < y, we have that

f(x) ≤ f(y). (4.11.7)

Proposition 4.11.8 Let f be a monotonically increasing real-valued function on (a, b). If p ∈ (a, b), then the one-sided limits of f at p exist, with respect to the standard Euclidean metric on R, and satisfy

f(p−) ≤ f(p) ≤ f(p+). (4.11.9)

If q is another element of (a, b), and p < q, then

f(p+) ≤ f(q−). (4.11.10)

Let p ∈ (a, b) be given, and put

A_{+}(p) = {f(x) : p < x < b} (4.11.11)

and A_{−}(p) = {f(x) : a < x < p}. (4.11.12)

Note that f(p) is a lower bound for A_{+}(p), and an upper bound for A_{−}(p), because f is monotonically increasing on (a, b). In this situation, the one-sided limits of f at p are given by

f(p+) = inf A_{+}(p) (4.11.13)

and f(p−) = sup A_{−}(p). (4.11.14)

To prove (4.11.13), let ϵ > 0 be given. Observe that there is a point x(ϵ) in (p, b) such that f(x(ϵ)) < inf A_{+}(p) + ϵ, (4.11.15)

by definition of the infimum. If p < x ≤ x(ϵ), then

inf A_{+}(p) ≤ f(x) ≤ f(x(ϵ)) < inf A_{+}(p) + ϵ, (4.11.16)

using the monotonicity of f in the second step. This implies (4.11.13), and the argument for (4.11.14) is analogous.

It is easy to obtain (4.11.9) from (4.11.13) and (4.11.14). If p < q < b, then let x be an element of (p, q). Using (4.11.13) and the analogue of (4.11.14) for q, we get that f(p+) ≤ f(x) ≤ f(q−). (4.11.17)

More precisely, this also uses the fact that f(x) is an element of A_{+}(p) and A_{−}(q). This implies (4.11.10).

Theorem 4.11.18 If f is a monotonically increasing real-valued function on (a, b), then f is continuous at all but finitely or countably many points in (a, b), with respect to the standard Euclidean metric on R and its restriction to (a, b).

Because the one-sided limits of f exist at every point in (a, b), f is continuous at p ∈ (a, b) if and only if

f(p+) = f(p−) = f(p). (4.11.19)

It follows that f is not continuous at p exactly when

f(p−) < f(p+), (4.11.20)

by (4.11.9). In this case, we can choose a rational number r(p) ∈ (f(p−), f(p+)). If f is not continuous at q ∈ (a, b) and p < q, then

r(p) < f(p+) ≤ f(q−) < r(q), (4.11.21)

by (4.11.10). Thus we get a one-to-one correspondence between the set of discontinuities of f in (a, b) and a set of rational numbers, which implies the theorem.

4.12 Prescribing jump discontinuities

Consider the real-valued function defined on the real line by

(4.12.1)  I(x) = 0 when x ≤ 0
= 1 when x > 0.

This may be called the unit step function on R, as in Definition 6.14 on p129 of [112]. Note that I(x) is monotonically increasing on R, and that it is discontinuous at 0.

If u ∈ R, then

(4.12.2)  I(x − u) = 0 when x ≤ u
= 1 when x > u

is monotonically increasing on R, and discontinuous at u. One can use linear combinations of these functions with positive coefficients to get monotonically increasing functions on R with discontinuities at any finite set of points.

Let {t_{j}}_{j=1}^{∞} be an infinite sequence of real numbers, and let {c_{j}}_{j=1}^{∞} be an infinite sequence of complex numbers such that

∑_{j=1}^{∞} |c_{j}| (4.12.3)

converges. Put

f(x) = ∑_{j=1}^{∞} c_{j} I(x − t_{j}) (4.12.4)

for each x ∈ R, where the series on the right converges absolutely, by the comparison test. If c_{j} is a nonnegative real number for each j, then it is easy to see that f is a monotonically increasing real-valued function on R.

The sequence of partial sums

∑_{j=1}^{n} c_{j} I(x − t_{j}) (4.12.5)

converges to f uniformly on R, by Weierstrass’ criterion, as in Section 4.6. If x ∈ R and x ≠ t_{j} for each j, then it follows that

(4.12.6) f is continuous at x,

as before. If the t_{j}’s are distinct elements of R, and c_{j} ≠ 0 for each j, then one can check that f is discontinuous at t_{l} for each l. More precisely, the one-sided limits of f at t_{l} exist for each l, and satisfy

f(t_{l}+) − f(t_{l}−) = c_{l} (4.12.7)

in this case.

This type of construction is described another way in Remark 4.31 on p97 of [112]. This description is mentioned in Theorem 6.16 on p130 of [112], and some of its properties are indicated in Exercise 8 on p166 of [112].

4.13 Path connectedness

Let (M, d(x, y)) be a metric space.

Proposition 4.13.1 Let a and b be real numbers with a < b, and let p be a continuous mapping from the closed interval [a, b] into M, with respect to the restriction of the standard Euclidean metric on R to [a, b]. Under these conditions, p([a, b]) is a connected subset of M.

As in an earlier proof, it is helpful to extend p to a mapping from the real line into M, by putting p(t) = p(a) when t < a, and p(t) = p(b) when t > b. It is easy to see that this extension is continuous as a mapping from R into M. It follows that p([a, b]) is a connected subset of M, because [a, b] is connected in R.

Definition 4.13.2 A subset E of M is said to be path connected if for every pair of points x, y ∈ E there is a continuous mapping p from the closed unit interval [0, 1] into M such that p(0) = x, p(1) = y, and p(t) ∈ E for every t ∈ [0, 1]. This uses the restriction of the standard Euclidean metric on R to [0, 1].

Proposition 4.13.3 If E ⊆ M is path connected, then E is connected in M.

Suppose for the sake of a contradiction that E is not connected in M, so that there are nonempty separated subsets A, B of M such that A∪B = E. Let x be an element of A, and let y be an element of B. If E is path connected in M, then there is a continuous mapping p from [0, 1] into M such that p(0) = x, p(1) = y, and p([0, 1]) ⊆ E.

Put A_{1} = A ∩ p([0, 1]), B_{1} = B ∩ p([0, 1]). (4.13.4)

Observe that A_{1} ∪ B_{1} = p([0, 1]), (4.13.5)

because p([0, 1]) ⊆ E = A ∪ B. By construction, x ∈ A_{1} and y ∈ B_{1}, so that A_{1}, B_{1} ≠ ∅. One can check that A_{1} and B_{1} are separated in M, because A and B are separated in M, and A_{1} ⊆ A, B_{1} ⊆ B. This implies that p([0, 1]) is not connected in M, which is a contradiction, as desired.

Definition 4.13.6 Let n be a positive integer. A subset E of R^{n} is said to be convex if for every x, y ∈ E and t ∈ R with 0 ≤ t ≤ 1, we have that

(1 − t) x + t y ∈ E. (4.13.7)

It is easy to see that convex subsets of R^{n} are path connected, with respect to the standard Euclidean metric on R^{n}. Note that connected subsets of the real line are convex.

Proposition 4.13.8 Let M and N be metric spaces, and let f be a continuous mapping from M into N. If E is a path-connected subset of M, then f(E) is path connected in N.

This follows from the definition of path connectedness, and the fact that compositions of continuous mappings are continuous.

It is well known and not too difficult to show that connected open subsets of R^{n} are connected. However, there are examples of connected subsets of R^{2} that are not path connected.

4.14 Integral metrics

In this section, the reader is supposed to be familiar with the Riemann integral of a continuous function on a closed interval in the real line. Let a, b be real numbers, with a < b, and let

C([a, b]) = C([a, b], R) (4.14.1)

be the space of continuous real-valued functions on [a, b]. More precisely, this uses the standard Euclidean metric on R, and its restriction to [a, b].

If f, g ∈ C([a, b]), then put

d_{1}(f, g) = ∫_{a}^{b} |f(x) − g(x)| dx. (4.14.2)

One can check that this defines a metric on C([a, b]). In particular, d_{1}(f, g) > 0 unless f = g on [a, b].

Remember that continuous real-valued functions on [a, b] are automatically bounded, because [a, b] is compact. The supremum metric on C([a, b]) is defined by θ(f, g) = sup{|f(x) − g(x)| : a ≤ x ≤ b}, (4.14.3)

as in Section 4.8. It is easy to see that

d_{1}(f, g) ≤ (b − a) θ(f, g) (4.14.4)

for every f, g ∈ C([a, b]).

In particular, if {f_{j}}_{j=1}^{∞} is a sequence of continuous real-valued functions on [a, b] that converges uniformly to a continuous real-valued function f on [a, b], then {f_{j}}_{j=1}^{∞} converges to f with respect to d_{1}(·, ·).

However, C([a, b]) is not complete as a metric space with respect to (4.14.2). To get a complete metric space, one can use the Lebesgue integral.

4.15 Some additional topics

Let (M, d(x, y)) and (N, ρ(u, v)) be metric spaces. A mapping ϕ from M into N is said to be an isometry if

ρ(ϕ(x), ϕ(y)) = d(x, y) (4.15.1)

for every x, y ∈ M. A completion of M may be defined as an isometry from M onto a dense subset of a complete metric space. It is well known that every metric space has a completion, which is unique up to isometric equivalence.

Suppose that M is complete. If U_{1}, U_{2}, U_{3}, … is an infinite sequence of dense open subsets of M, then the Baire category theorem states that ⋂_{j=1}^{∞} U_{j} is dense in M too.

A mapping ϕ from M into itself is said to be a contraction if there is a nonnegative real number c < 1 such that

d(ϕ(x), ϕ(y)) ≤ c d(x, y) (4.15.2)

for every x, y ∈ M. In this case, if M is also nonempty and complete, then the contraction mapping theorem states that there is a unique w ∈ M such that ϕ(w) = w.

Let n be a positive integer, and let

B_{n} = {x ∈ R^{n} : ∑_{j=1}^{n} x_{j}^{2} ≤ 1} (4.15.3)

be the closed unit ball in R^{n}. Suppose that f is a continuous mapping from B_{n} into itself, with respect to the restriction of the standard Euclidean metric on R^{n} to B_{n}. Brouwer’s fixed-point theorem says that there is an x ∈ B_{n} such that f(x) = x. This can be obtained from the intermediate value theorem when n = 1.

Let a and b be real numbers, with a < b. Suppose that f is a continuous real-valued function on [a, b], with respect to the standard Euclidean metric on R and its restriction to [a, b]. Under these conditions, a famous theorem of Weierstrass says that f can be uniformly approximated by polynomials on [a, b].

Some derivatives and integrals

5.1 Functions on subsets of R

Let E be a nonempty subset of the real line, and let f be a real-valued function on E. In this and the next sections, we always use the standard Euclidean metric on R, and its restriction to E. If

(5.1.1) x ∈ E is a limit point of E,

then the derivative of f at x as a function on E may be defined as usual by

f′(x) = lim_{{w∈E \atop w→x}} \frac{f(w) − f(x)}{w − x}, (5.1.2)

when this limit exists. In this case, f is said to be differentiable at x, as a function on E.

Of course, one is often particularly interested in functions defined on something like an interval in the real line. Some of the usual arguments work for other types of subsets of R, and some arguments involve more properties of E. See [76, 77] for some topics related to functions defined on rational numbers.

If f is differentiable at x as a function on E, then

(5.1.3) f is continuous at x as a function on E.

Indeed, under these conditions, we have that

lim_{{w∈E \atop w→x}}(f(w) − f(x)) = lim_{{w∈E \atop w→x}} ((\frac{f(w) − f(x)}{w − x}) (w − x))

(5.1.4)  = ( lim_{{w∈E \atop w→x}} (\frac{f(w) − f(x)}{w − x})) ( lim_{{w∈E \atop w→x}}(w − x))
= f′(x) · 0 = 0,

where the second step is as in Section 4.10.

Let g be another real-valued function on E that is differentiable at x. It is easy to see that

(5.1.5) f + g is differentiable at x,

with

(f + g)′(x) = f′(x) + g′(x). (5.1.6)

This uses the analogous statement for limits of sums in Section 4.10. One can also check that

(5.1.7) f g is differentiable at x,

with

(f g)′(x) = f′(x) g(x) + f(x) g′(x), (5.1.8)

which is the usual product rule for the derivative. This can be obtained in a standard way from the usual results about limits of products in Section 4.10.

Suppose that

f(w) ≠ 0 (5.1.9)

for each w ∈ E, so that

1/f (5.1.10)

defines a real-valued function on E too. In this case, one can show that

(5.1.11) 1/f is differentiable at x,

with

(1/f)′(x) = −f′(x)/f(x)^{2}. (5.1.12)

This uses the fact that 1/f is continuous at x as a function on E, because of (5.1.3). Note that (5.1.12) could be obtained from the product rule, if one knows (5.1.11).

More precisely, one can consider 1/f as a real-valued function on the set

{w ∈ E : f(w) ≠ 0}, (5.1.13)

even if f(w) = 0 for some w ∈ E. If f(x) ≠ 0, then (5.1.13) contains every w ∈ E that is sufficiently close to x, because f is continuous at x. In particular, this means that x is a limit point of (5.1.13), because x is a limit point of E in R, by hypothesis.

Of course, the quotient rule for the derivative can be obtained from (5.1.12) and the product rule.

82 CHAPTER 5. SOME DERIVATIVES AND INTEGRALS

5.2 The chain rule

Let E be a nonempty subset of the real line again, and let f be a real-valued function on E. Suppose that x ∈ E is a limit point of E, and that f is differentiable at x. Let E_{1} be another subset of the real line, and suppose that f(E) ⊆ E_{1}, (5.2.1) and that (5.2.2) f(x) is a limit point of E_{1}. Also let g be a real-valued function on E_{1}, and put h = g ∘ f, (5.2.3) which is a real-valued function on E. If (5.2.4) g is differentiable at f(x), then the chain rule states that (5.2.5) h is differentiable at x, with h′(x) = g′(f(x)) f′(x). (5.2.6) This is very easy to see when g is a linear function on R. Using this, one can reduce to the case where g′(f(x)) = 0, (5.2.7) which is a bit simpler.

Suppose for the moment that f is a one-to-one function on E, and that g is the inverse of f on E_{1} = f(E), (5.2.8) so that h(w) = g(f(w)) = w (5.2.9) for every w ∈ E. It is easy to see that (5.2.2) holds in this case, because f is continuous at x, as in the previous section. If (5.2.4) holds, then g′(f(x)) f′(x) = h′(x) = 1. (5.2.10) This means that f′(x) ≠ 0, and that g′(f(x)) = 1/f′(x). (5.2.11)

Suppose for the moment again that E = (a, b) (5.2.12) for some real numbers a, b with a < b, although one could also permit a = −∞ or b = +∞. Suppose also that f is differentiable at every w ∈ (a, b), with

f′(w) > 0. (5.2.13)

One can use the mean value theorem to get that f is strictly increasing on (a, b). One can also check that f((a, b)) is an open interval in R, or an open half-line, or R. If g is the inverse of f on f((a, b)), then part of Exercise 2 on p114 of [112] states that

(5.2.14) g is differentiable at every point in f((a, b)).

Let E be any nonempty subset of R again, and suppose that x ∈ E is a limit point of E, as before. Suppose that f is one-to-one on E, and let g be the inverse of f on (5.2.8). Suppose also that f is differentiable at x, as usual, with f′(x) ≠ 0. If (5.2.15) g is continuous at f(x),

as a real-valued function on (5.2.8), then one can show that (5.2.4) holds. This corresponds to Theorem 7.5 E on p174 of [47], at least when E is an interval in R, which may be unbounded, as well as open, closed, or half-open and halfclosed, as on p2 of [47].

More precisely, (5.2.15) implies that if w ∈ E and f(w) is close to f(x), then

(5.2.16) w = g(f(w)) is close to x = g(f(x)).

Otherwise, let r_{0} be a positive real number, and let g_{0} be the restriction of g to

f(E ∩ (x − r_{0}, x + r_{0})). (5.2.17)

If r_{0} is small enough, then one can check that g_{0} is continuous at f(x) as a function on (5.2.17), using the hypothesis that f′(x) ≠ 0.

5.3 One-sided derivatives

Let E be a nonempty subset of R again, and let f be a real-valued function on E. If x ∈ E, then put E^{+}(x) = {w ∈ E : w ≥ x} (5.3.1)

and E^{−}(x) = {w ∈ E : w ≤ x}. (5.3.2)

If (5.3.3) x is a limit point of E^{+}(x),

then the derivative of f at x from the right as a function on E is defined by

f′_{+}(x) = lim_{{w∈E^{+}(x) \atop w→x}} \frac{f(w) − f(x)}{w − x}, (5.3.4)

when this limit exists.

Similarly, if (5.3.5) x is a limit point of E^{−}(x),

then the derivative of f at x from the left as a function on E is defined by

f′_{−}(x) = lim_{{w∈E^{−}(x) \atop w→x}} \frac{f(w) − f(x)}{w − x}, (5.3.6)

when this limit exists. These are the same as the derivatives at x of the restrictions of f to E^{+}(x) and E^{−}(x), respectively, when they exist.

If x is a limit point of E^{+}(x) or E^{−}(x), then it is easy to see that x is a limit point of E. One can check that the converse holds as well.

Suppose for the moment that

(5.3.7) x is a limit point of E^{+}(x) and E^{−}(x),

so that x is a limit point of E in particular. If f is differentiable at x as a function on E, then it is easy to see that the derivatives of f at x from the left and right exist, with f′_{+}(x) = f′_{−}(x) = f′(x). (5.3.8)

Conversely, if the derivatives of f at x from the left and right exist and are equal to each other, then f is differentiable at x as a function on E, with the derivative of f at x as in (5.3.8). This is very similar to a remark in Section 4.11.

5.4 More on one-sided derivatives

Let us continue with the same notation and hypotheses as in the previous section. Suppose for the moment again that

(5.4.1) f has a local maximum at x,

as a function on E, so that f(w) ≤ f(x) (5.4.2)

for all w ∈ E that are sufficiently close to x. If x is a limit point of E^{+}(x), and if the derivative of f at x from the right as a function on E exists, then it is easy to see that f′_{+}(x) ≤ 0. (5.4.3)

Similarly, if x is a limit point of E^{−}(x), and if the derivative of f at x from the left as a function on E exists, then

f′_{−}(x) ≥ 0. (5.4.4)

If (5.3.7) holds, and if f is differentiable at x as a function on E, then it follows that f′(x) = 0, (5.4.5) because of (5.3.8).

Similarly, suppose for the moment that

(5.4.6) f has a local minimum at x,

as a function on E, so that f(x) ≤ f(w) (5.4.7)

for all w ∈ E that are sufficiently close to x. If x is a limit point of E^{+}(x), and if the derivative of f at x from the right as a function on E exists, then

f′_{+}(x) ≥ 0. (5.4.8)

If x is a limit point of E^{−}(x), and if the deriviative of f at x from the left as a function on E exists, then f′_{−}(x) ≤ 0. (5.4.9)

If (5.3.7) holds, and if f is differentiable at x as a function on E, then (5.4.5) holds, because of (5.3.8), as before. Of course, these statements correspond to those in the preceding paragraph for −f.

These remarks are related to Rolle’s theorem and the mean value theorem when E is a closed interval in R. Remember that Rolle’s theorem uses the extreme value theorem, as in Corollary 4.3.9. One might consider analogous types of arguments when E is the Cantor set, as in Section 2.9, for instance.

See [6, 24, 27, 75, 123] for some additional topics concerning the mean value theorem. Some related matters are discussed in [34, 36].

Let a and b be real numbers with a < b, and let f, g be continuous realvalued functions on [a, b] that are differentiable at every point in (a, b). Under these conditions, the generalized mean value theorem states that there is an x ∈ (a, b) such that

(f(b) − f(a)) g′(x) = (g(b) − g(a)) f′(x), (5.4.10)

as in Theorem 5.9 on p107 of [112]. To see this, put

h(t) = (f(b) − f(a)) g(t) − (g(b) − g(a)) f(t) (5.4.11)

for every t ∈ [a, b]. This is a continuous real-valued function on [a, b] that is differentiable at every point in (a, b), with

h(a) = f(b) g(a) − f(a) g(b) = h(b). (5.4.12)

One can use Rolle’s theorem to get a point x ∈ (a, b) such that h′(x) = 0, and (5.4.10) follows from this.

5.5 The Darboux property

Let a and b be real numbers with a < b, and let f be a real-valued function on [a, b] that is differentiable at every point in [a, b]. Suppose that λ ∈ R satisfies

f′(a) < λ < f′(b) (5.5.1) or

f′(b) < λ < f′(a). (5.5.2)

Under these conditions, there is an x ∈ (a, b) such that

f′(x) = λ. (5.5.3)

To see this, it is convenient to suppose that (5.5.1) holds, and otherwise one can use an analogous argument when (5.5.2) holds, or reduce to this case. Let g be the real-valued function defined on [a, b] by

g(w) = f(w) − λ w, (5.5.4)

and observe that g is differentiable on [a, b], with

g′ = f′ − λ. (5.5.5)

In particular,

g′(a) = f′(a) − λ < 0 (5.5.6)

and

g′(b) = f′(b) − λ > 0, (5.5.7)

because of (5.5.1).

The extreme value theorem implies that g attains its minimum at a point x ∈ [a, b]. One can check that x ∈ (a, b) in this case, using (5.5.6) and (5.5.7), as in the previous section. It follows that

g′(x) = 0, (5.5.8)

as in the previous section again. This implies (5.5.3), by (5.5.5). Of course, the same conclusion can be obtained from the intermediate value theorem when f′ is continuous on [a, b].

Let f be a real-valued function on the real line that is differentiable at every point in R. Put

f_{j}(x) = j (f(x + (1/j)) − f(x)) (5.5.9)

for every x ∈ R and positive integer j, and observe that

lim_{j→∞} f_{j}(x) = f′(x) (5.5.10)

for every x ∈ R. Pointwise limits of continuous functions are said to be of Baire class one. Thus f′ is of Baire class one. Some related references include [12, 19, 20, 21, 31].

5.6 Some examples

Let l be a positive integer, and consider the function f_{l} defined on the real line by

(5.6.1)  f_{l}(x) = x^{2} sin(x^{−l}) when x ≠ 0
= 0  when x = 0.

We shall use standard properties of trigonometric functions here, although one could avoid this by using other nonconstant periodic differentiable functions on R. If x ≠ 0, then f_{l} is differentiable at x, with

(5.6.2)  f′_{l}(x) = 2 x sin(x^{−l}) + x^{2} (−l x^{−l−1}) cos(x^{−l})
= 2 x sin(x^{−l}) − l x^{1−l} cos(x^{−l}).

One can check directly that f_{l} is differentiable at 0, with

f′_{l}(0) = 0. (5.6.3)

Note that

lim_{x→0}(2 x sin(x^{−l})) = 0, (5.6.4)

because sin(x^{−l}) is bounded for x ≠ 0. If l = 1, then (5.6.2) reduces to

f′_{1}(x) = 2 x sin(1/x) − cos(1/x) (5.6.5)

for x ≠ 0. It is easy to see that f′_{1} is not continuous at 0. If l > 1, then f′_{l}(x) is not even bounded for x ≠ 0 near 0.

Clearly (5.6.5) is bounded on bounded subsets of R \ {0}. Of course, it is well known that

(5.6.6)  lim_{t→0}(t^{−1} sin(t)) = sin′(0) = 1.

One can use this with t = 1/x to get that

f′_{1}(x) → 1 as x → ±∞. (5.6.7)

In particular, this implies that (5.6.5) is bounded on R \ {0}. Similarly, if l > 1, then one can use (5.6.6) with t = x^{−l} to get that

f′_{l}(x) → 0 as x → ±∞. (5.6.8)

See [72] for more on some related examples, and their properties.

5.7 Lipschitz mappings

Let (M, d(·, ·)) and (N, ρ(·, ·)) be metric spaces. A mapping f from M into N is said to be Lipschitz if there is a nonnegative real number C such that

ρ(f(x), f(w)) ≤ C d(x, w) (5.7.1) for all x, w ∈ M. In this case, we may say that f is Lipschitz with constant C, to indicate the role of C. It is easy to see that

(5.7.2) Lipschitz mappings are uniformly continuous.

Note that

(5.7.3) f is Lipschitz with constant C = 0 on M if and only if f is constant on M.

Let E be a subset of the real line, and let f be a real-valued function on E. Note that f is Lipschitz with constant C ≥ 0 on E exactly when

|f(x) − f(w)| ≤ C |x − w| (5.7.4)

for all x, w ∈ E. This uses the standard Euclidean metric on R, and its restriction to E.

Suppose that x ∈ E is a limit point of E, and that f is differentiable at x. If f is Lipschitz with constant C on E, then one can check that

|f′(x)| ≤ C. (5.7.5)

Let a and b be real numbers with a < b, and let f be a continuous real-valued function on [a, b]. Suppose that f is differentiable at every point in (a, b), and that there is a nonnegative real number C such that (5.7.5) holds for every x ∈ (a, b). Under these conditions, one can verify that

(5.7.6) f is Lipschitz with constant C on [a, b],

using the mean value theorem.

Let E be a subset of the real line again, and let f be a real-valued function on E. Suppose that x_{0} ∈ E is a limit point of E, and that f is differentiable at x_{0}. If ϵ_{0} is a positive real number, then one can check that there is a positive real number δ_{0} such that

|\frac{f(w) − f(x_{0})}{w − x_{0}} | < |f′(x_{0})| + ϵ_{0} (5.7.7)

for every w ∈ E with w ≠ x_{0} and |w − x_{0}| < δ_{0}. This implies that

|f(w) − f(x_{0})| ≤ (|f′(x_{0})| + ϵ_{0}) |w − x_{0}| (5.7.8)

for every w ∈ E with |w −x_{0}| < δ_{0}, because (5.7.8) holds trivially when w = x_{0}. This may be considered as a type of pointwise Lipschitz condition for f at x_{0}. In particular, this implies that f is continuous at x_{0}. Remember that the continuity of f at x_{0} was also mentioned in Section 5.1.

5.8. SUPREMA AND INFIMA OF FUNCTIONS 89

5.8 Suprema and infima of functions

Let E be a nonempty set, and let f be a real-valued function on E. Suppose that f is bounded on E, so that f(E) is a nonempty bounded subset of the real line. Under these conditions, we may use the notation (5.8.1) sup_{x∈E} f(x) = sup f(E) = sup{f(x) : x ∈ E} and (5.8.2) inf_{x∈E} f(x) = inf f(E) = inf{f(x) : x ∈ E}. Of course,

inf_{x∈E} f(x) ≤ sup_{x∈E} f(x). (5.8.3)

If t ∈ R, then it is easy to see that (5.8.4) t f is a bounded real-valued function on E too. If t > 0, then one can check that

sup_{x∈E}(t f(x)) = t ( sup_{x∈E} f(x)) (5.8.5)

and

inf_{x∈E}(t f(x)) = t ( inf_{x∈E} f(x)). (5.8.6)

Similarly, if t < 0, then

sup_{x∈E}(t f(x)) = t ( inf_{x∈E} f(x)) (5.8.7)

and

inf_{x∈E}(t f(x)) = t ( sup_{x∈E} f(x)). (5.8.8)

Let g be another bounded real-valued function on E, and observe that (5.8.9) f + g is bounded on E as well. One can verify that (5.8.10) sup_{x∈E}(f(x) + g(x)) ≤ ( sup_{x∈E} f(x)) + ( sup_{x∈E} g(x)) and

(5.8.11)  inf_{x∈E}(f(x) + g(x)) ≥ ( inf_{x∈E} f(x)) + ( inf_{x∈E} g(x)).

90 CHAPTER 5. SOME DERIVATIVES AND INTEGRALS

5.9 More on suprema and infima of functions

Let us continue with the same notation and hypotheses as in the previous section. We also have that (5.9.1) sup_{x∈E}(f(x) + g(x)) ≥ ( sup_{x∈E} f(x)) + ( inf_{x∈E} g(x)) and (5.9.2) inf_{x∈E}(f(x) + g(x)) ≤ ( inf_{x∈E} f(x)) + ( sup_{x∈E} g(x)). This can be verified directly, or using the previous inequalities and some of the earlier remarks with t = −1.

Suppose for the moment that f, g ≥ 0 on E. (5.9.3) Under these conditions, one can check that (5.9.4) sup_{x∈E}(f(x) g(x)) ≤ ( sup_{x∈E} f(x)) ( sup_{x∈E} g(x)) and (5.9.5) inf_{x∈E}(f(x) g(x)) ≥ ( inf_{x∈E} f(x)) ( inf_{x∈E} g(x)).

Suppose for the moment again that f(x) ≤ g(x) (5.9.6) for every x ∈ E. In this case, it is easy to see that

sup_{x∈E} f(x) ≤ sup_{x∈E} g(x) (5.9.7)

and

inf_{x∈E} f(x) ≤ inf_{x∈E} g(x). (5.9.8)

If E_{0} is a nonempty subset of E, then

sup_{x∈E_{0}} f(x) ≤ sup_{x∈E} f(x) (5.9.9)

and

inf_{x∈E} f(x) ≤ inf_{x∈E_{0}} f(x). (5.9.10)

5.10 Darboux sums

Let a and b be real numbers, with a ≤ b, and let P = {t_{j}}_{j=0}^{l} (5.10.1) be a partition of [a, b]. More precisely, this is a finite sequence of real numbers with a = t_{0} ≤ t_{1} ≤ ⋯ ≤ t_{l−1} ≤ t_{l} = b. (5.10.2)

It is convenient to include the case where a = b here, but the various sums and integrals that we shall consider will be equal to 0 when this happens. Similarly, if t_{j−1} = t_{j} for some j ≥ 1, then the corresponding term in each of the sums that we shall consider will be equal to 0.

Let f be a bounded real-valued function on [a, b], and put

M_{j} = sup{f(x) : t_{j−1} ≤ x ≤ t_{j}} (5.10.3)

and m_{j} = inf{f(x) : t_{j−1} ≤ x ≤ t_{j}} (5.10.4)

for each j = 1, …, l. Consider

U(P, f) = ∑_{j=1}^{l} M_{j} (t_{j} − t_{j−1}) (5.10.5)

and

L(P, f) = ∑_{j=1}^{l} m_{j} (t_{j} − t_{j−1}), (5.10.6)

as on p121 of [112], for instance. These are the upper and lower Darboux sums associated to f and P, respectively.

If w_{j} ∈ [t_{j−1}, t_{j}], 1 ≤ j ≤ l, (5.10.7)

then

L(P, f) ≤ ∑_{j=1}^{l} f(w_{j}) (t_{j} − t_{j−1}) ≤ U(P, f). (5.10.8)

The sum in the middle is the Riemann sum associated to f, w_{1}, …, w_{l}, and the partition P.

5.11 Darboux–Stieltjes sums

Let us continue with the same notation and hypotheses as in the previous section. Also let α be a monotonically increasing real-valued function on [a, b], so that α(t_{j}) − α(t_{j−1}) ≥ 0 (5.11.1)

for each j = 1, …, l. Consider

U(P, f, α) = ∑_{j=1}^{l} M_{j} (α(t_{j}) − α(t_{j−1})) (5.11.2)

and

L(P, f, α) = ∑_{j=1}^{l} m_{j} (α(t_{j}) − α(t_{j−1})), (5.11.3)

as on p122 of [112]. These are the upper and lower Darboux sums or Darboux– Stieltjes sums associated to f, P, and α, respectively.

If w_{1}, …, w_{l} are as in (5.10.7), then

L(P, f, α) ≤ ∑_{j=1}^{l} f(w_{j}) (α(t_{j}) − α(t_{j−1})) ≤ U(P, f, α). (5.11.4)

The sum in the middle is the Riemann–Stieltjes sum associated to f, the points w_{1}, …, w_{l}, the partition P, and α.

Of course, if α(x) = x on [a, b], then (5.11.2) and (5.11.3) are the same as (5.10.5) and (5.10.6), respectively. Note that

∑_{j=1}^{l}(α(t_{j}) − α(t_{j−1})) = ∑_{j=1}^{l} α(t_{j}) − ∑_{j=1}^{l} α(t_{j−1})

(5.11.5)  = ∑_{j=1}^{l} α(t_{j}) − ∑_{j=0}^{l−1} α(t_{j}) = α(b) − α(a).

5.12 More on Darboux sums

Let us continue with the same notation and hypotheses as in the previous two sections. Clearly m_{j} ≤ M_{j} (5.12.1)

for each j. This implies that

L(P, f, α) ≤ U(P, f, α), (5.12.2)

which could also be obtained from (5.11.4). In particular,

L(P, f) ≤ U(P, f), (5.12.3)

which follows from (5.10.8) as well.

If we put M = sup{f(x) : a ≤ x ≤ b} (5.12.4)

and m = inf{f(x) : a ≤ x ≤ b}, (5.12.5)

then

M = max_{1≤j≤l} M_{j} (5.12.6)

and

m = min_{1≤j≤l} m_{j}. (5.12.7)

Using this, we get that

U(P, f, α) ≤ ∑_{j=1}^{l} M (α(t_{j}) − α(t_{j−1})) = M (α(b) − α(a)) (5.12.8)

and

L(P, f, α) ≥ ∑_{j=1}^{l} m (α(t_{j}) − α(t_{j−1})) = m (α(b) − α(a)). (5.12.9)

In particular, U(P, f) ≤ M (b − a) (5.12.10)

and L(P, f) ≥ m (b − a), (5.12.11)

as on p121 of [112]. Note that equality holds in these four inequalities when P is the partition of [a, b] consisting only of a and b, so that l = 1.

Let P^{∗} = {t_{k}^{∗}}_{k=0}^{l^{∗}} be another partition of [a, b]. We say that

(5.12.12) P^{∗} is a refinement of P

if for each j = 0, 1 …, l there is a k ∈ {0, 1, …, l^{∗}} such that

t_{j} = t_{k}^{∗}. (5.12.13)

This means that [t_{j−1}, t_{j}] may be partitioned further by P^{∗} for each j = 1, …, l. In this case, one can show that

U(P^{∗}, f, α) ≤ U(P, f, α) (5.12.14)

and L(P, f, α) ≤ L(P^{∗}, f, α), (5.12.15)

as in Theorem 6.4 on p123 of [112].

5.13 Upper and lower integrals

Let us continue with the same notation and hypotheses as in the previous three sections. The upper and lower Riemann integrals of f on [a, b] are defined by

\overline{∫} _{a}^{b}f dx = inf_{P} U(P, f) (5.13.1)

and

\underline{∫} _{a}^{b}f dx = sup_{P} L(P, f), (5.13.2)

respectively, as on p121 of [112]. More precisely, this infimum and supremum are taken over all partitions P of [a, b].

Remember that

m (b − a) ≤ L(P, f) ≤ U(P, f) ≤ M (b − a) (5.13.3)

for all such partitions P, as in (5.12.3), (5.12.10), and (5.12.11), where M and m are as in (5.12.4) and (5.12.5), respectively. This implies that the infimum and supremum in (5.13.1) and (5.13.2) exist in R, with

m (b − a) ≤ \overline{∫} _{a}^{b}f dx (5.13.4)

and

\underline{∫} _{a}^{b}f dx ≤ M (b − a). (5.13.5)

We also have that

\overline{∫} _{a}^{b}f dx ≤ M (b − a) (5.13.6)

and

\underline{∫} _{a}^{b}f dx ≥ m (b − a). (5.13.7)

Similarly, the upper and lower Riemann–Stieltjes integrals of f with respect to α on [a, b] are defined by

\overline{∫} _{a}^{b}f dα = inf_{P} U(P, f, α) (5.13.8)

and

\underline{∫} _{a}^{b}f dα = sup_{P} L(P, f, α), (5.13.9)

respectively, as on p122 of [112]. These are the same as in (5.13.1) and (5.13.2) when α(x) = x on [a, b].

Note that

m (α(b) − α(a)) ≤ L(P, f, α) ≤ U(P, f, α) ≤ M (α(b) − α(a)) (5.13.10)

for all partitions P of [a, b], as in (5.12.2), (5.12.8), and (5.12.9). This implies that the infimum and supremum in (5.13.8) and (5.13.9) exist in R, with

m (α(b) − α(a)) ≤ \overline{∫} _{a}^{b}f dα (5.13.11)

and

\underline{∫} _{a}^{b}f dα ≤ M (α(b) − α(a)). (5.13.12)

In addition,

\overline{∫} _{a}^{b}f dα ≤ M (α(b) − α(a)) (5.13.13)

and

\underline{∫} _{a}^{b}f dα ≥ m (α(b) − α(a)). (5.13.14)

5.14 Common refinements

Let us continue with the same notation and hypotheses as in the previous four sections. Let P_{1} and P_{2} be any two partitions of [a, b], as in the previous section. It is not too difficult to see that there is a partition P^{∗} of [a, b] such that

(5.14.1) P^{∗} is a refinement of both P_{1}, P_{2}.

The smallest such common refinement of P_{1} and P_{2} can be obtained using all of the points in [a, b] that occur in P_{1} and P_{2}, listed in order.

Under these conditions, we have that

L(P_{1}, f, α) ≤ L(P^{∗}, f, α) ≤ U(P^{∗}, f, α) ≤ U(P_{2}, f, α). (5.14.2)

This uses (5.12.15) in the first step, (5.12.2) in the second step, and (5.12.14) in the third step. One can use this to get that

\underline{∫} _{a}^{b}f dα ≤ \overline{∫} _{a}^{b}f dα, (5.14.3)

as in Theorem 6.5 on p124 of [112]. In particular,

\underline{∫} _{a}^{b}f dx ≤ \overline{∫} _{a}^{b}f dx. (5.14.4)

If P = {t_{j}}_{j=0}^{l} is a partition of [a, b], then

U(P, f, α) − L(P, f, α) = ∑_{j=1}^{l}(M_{j} − m_{j}) (α(t_{j}) − α(t_{j−1})), (5.14.5)

by (5.11.2) and (5.11.3). Remember that M_{j} and m_{j} are as in (5.10.3) and (5.10.4), respectively, so that

M_{j} − m_{j} ≥ 0 (5.14.6)

for each j in particular.

Let P, P^{∗} be partitions of [a, b], and suppose that P^{∗} is a refinement of P. Observe that

U(P^{∗}, f, α) − L(P^{∗}, f, α) ≤ U(P, f, α) − L(P, f, α), (5.14.7)

because of (5.12.14) and (5.12.15). This is related to part (a) of Theorem 6.7 on p125 of [112], and its proof.

5.15 Riemann–Stieltjes integrals

We continue with the same notation and hypotheses as in the previous sections. If

\underline{∫} _{a}^{b}f dx = \overline{∫} _{a}^{b}f dx, (5.15.1)

then f is said to be Riemann integrable on [a, b], as on p121 of [112]. In this case, the Riemann integral of f on [a, b] is denoted

∫_{a}^{b} f dx, (5.15.2)

and defined to be the common value in (5.15.1). The Riemann integral may also be expressed as

∫_{a}^{b} f(x) dx. (5.15.3)

Similarly, if

\underline{∫} _{a}^{b}f dα = \overline{∫} _{a}^{b}f dα, (5.15.4)

then f is said to be Riemann–Stieltjes integrable with respect to α on [a, b], as on p122 of [112]. The corresponding Riemann–Stieltjes integral

∫_{a}^{b} f dα (5.15.5)

is defined to be the common value in (5.15.4), and this may also be expressed as

∫_{a}^{b} f(x) dα(x). (5.15.6)

This is the same as in the previous paragraph when α(x) = x on [a, b], as usual.

One can show that f is Riemann–Stieltjes integrable with respect to α on [a, b] if and only if for every ϵ > 0 there is a partition P of [a, b] such that

U(P, f, α) − L(P, f, α) < ϵ, (5.15.7)

as in Theorem 6.6 on p124 of [112]. More precisely, it is somewhat easier to see that f is Riemann–Stieltjes integrable with respect to α on [a, b] if and only if for every ϵ > 0 there are partitions P_{1}, P_{2} of [a, b] such that

U(P_{2}, f, α) − L(P_{1}, f, α) < ϵ. (5.15.8)

In order to get (5.15.7) from (5.15.8), one can take P to be a common refinement of P_{1} and P_{2}, and use (5.12.14), (5.12.15).

5.16 Continuity and integrability

Let us continue with the same notation and hypotheses as in the previous sections. If (5.16.1) f is continuous on [a, b],

then one can show that

(5.16.2) f is Riemann–Stieltjes integrable with respect to α on [a, b],

as in Theorem 6.8 on p125 of [112]. This uses the fact that f is uniformly continuous on [a, b], because [a, b] is compact. If c is a real number, then c may be considered as a constant function on [a, b], and it is easy to see that

∫_{a}^{b} c dα = c (α(b) − α(a)). (5.16.3)

If (5.16.4) f is continuous at all but finitely many points in [a, b],

and if (5.16.5) α is continuous at those points,

then one can show that (5.16.2) holds, as in Theorem 6.10 on p126 of [112]. If

(5.16.6) f is monotonic on [a, b],

and (5.16.7) α is continuous on [a, b],

then (5.16.2) holds, as in Theorem 6.9 on p126 of [112].

If

(5.16.8)  f(x) = 0 when x ∈ [a, b] is irrational
= 1 when x ∈ [a, b] is rational,

then can check that

\overline{∫} _{a}^{b}f dα = α(b) − α(a) (5.16.9)

and

\underline{∫} _{a}^{b}f dα = 0. (5.16.10)

This implies that f is not Riemann–Stieltjes integrable with respect to α on [a, b] when α is not constant on [a, b]. In particular, f is not Riemann integrable on [a, b] when a < b, as in Exercise 4 on p138 of [112].

5.17 More on Riemann–Stieltjes integrals

We continue with the same notation and hypotheses as in the previous sections. Let I(x) be the unit step function on the real line, as in Section 4.12. Also let w be a real number with a ≤ w < b, (5.17.1)

and put α_{w}(x) = I(x − w) (5.17.2)

for x ∈ [a, b], which is a monotonically increasing real-valued function on [a, b]. If (5.17.3) f is continuous at w,

then (5.16.2) holds, with α = α_{w}, and

∫_{a}^{b} f dα_{w} = f(w). (5.17.4)

This corresponds to Theorem 6.15 on p130 of [112], which is stated for w ≠ a.

The argument for w = a is very similar, but if w = b, then α_{w} ≡ 0 on [a, b], because of the way that I(x) is defined. Some related results are mentioned in Exercise 3 on p138 of [112]. In particular, one can consider functions defined on R in the same way as I(x) when x ≠ 0, and equal to 1 or 1/2 at x = 0.

Suppose that (5.16.2) holds, and let P = {t_{j}}_{j=0}^{l} be a partition of [a, b]. Of course,

L(P, f, α) ≤ ∫_{a}^{b} f dα ≤ U(P, f, α), (5.17.5)

by definition of the integral. If w_{j} ∈ [t_{j−1}, t_{j}], 1 ≤ j ≤ l, then one can use (5.11.4) and (5.17.5) to get that

|∑_{j=1}^{l} f(w_{j}) (α(t_{j}) − α(t_{j−1})) − ∫_{a}^{b} f dα| ≤ U(P, f, α) − L(P, f, α). (5.17.6)

In particular, if (5.15.7) holds for some ϵ > 0, then

|∑_{j=1}^{l} f(w_{j}) (α(t_{j}) − α(t_{j−1})) − ∫_{a}^{b} f dα| < ϵ. (5.17.7)

This corresponds to part (c) of Theorem 6.7 on p125 of [112].

5.18 Multiplying functions by constants

Let us continue with the same notation and hypotheses as in the previous sections again. Let r be a positive real number, so that

(5.18.1) r f is a bounded real-valued function on [a, b].

5.18. MULTIPLYING FUNCTIONS BY CONSTANTS 99 If P is a partition of [a, b], then one can check that U(P, r f, α) = r U(P, f, α) (5.18.2) and L(P, r f, α) = r L(P, f, α), (5.18.3) using some remarks in Section 5.8. This implies that

\overline{∫} _{a}^{b}r f dα = r \overline{∫} _{a}^{b}f dα (5.18.4) and \underline{∫} _{a}^{b}r f dα = r \underline{∫} _{a}^{b}f dα, (5.18.5) using the same remarks from Section 5.8. If f is Riemann–Stieltjes integrable with respect to α on [a, b], then it follows that (5.18.6) r f is Riemann–Stieltjes integrable with respect to α on [a, b] too, with ∫_{a}^{b} r f dα = r ∫_{a}^{b} f dα, (5.18.7) as in part (a) of Theorem 6.12 on p128 of [112].

Of course, (5.18.8) −f is a bounded real-valued function on [a, b] as well. If P is a partition of [a, b], then one can verify that U(P, −f, α) = −L(P, f, α) (5.18.9) and L(P, −f, α) = −U(P, f, α), (5.18.10) using some other remarks in Section 5.8. This means that

\overline{∫} _{a}^{b} − f dα = −\underline{∫} _{a}^{b}f dα (5.18.11)

and \underline{∫} _{a}^{b} − f dα = −\overline{∫} _{a}^{b}f dα, (5.18.12) using these same remarks from Section 5.8. If f is Riemann–Stieltjes integrable with respect to α on [a, b], then we get that (5.18.13) −f is Riemann–Stieltjes integrable with respect to α on [a, b], 100 CHAPTER 5. SOME DERIVATIVES AND INTEGRALS with ∫_{a}^{b} −f dα = − ∫_{a}^{b} f dα. (5.18.14)

Let r be a positive real number again, and note that (5.18.15) r α is a monotonically increasing real-valued function on [a, b]. If P is a partition of [a, b], then it is easy to see that U(P, f, r α) = r U(P, f, α) (5.18.16) and L(P, f, r α) = r L(P, f, α). (5.18.17) This implies that \overline{∫} _{a}^{b}f d(r α) = r \overline{∫} _{a}^{b}f dα (5.18.18) and \underline{∫} _{a}^{b}f d(r α) = r \underline{∫} _{a}^{b}f dα, (5.18.19) using some remarks from Section 5.8, as before. If f is Riemann–Stieltjes integrable with respect to α on [a, b], then we get that (5.18.20) f is Riemann–Stieltjes integrable with respect to r α on [a, b], with ∫_{a}^{b} f d(r α) = r ∫_{a}^{b} f dα. (5.18.21) This corresponds to part of part (e) of Theorem 6.12 on p128 of [112].

5.19 Adding two functions

We continue with the same notation and hypotheses as in the previous sections. Let g be another bounded real-valued function on [a, b], so that (5.19.1) f + g is bounded on [a, b] as well. If P is a partition of [a, b], then one can check that U(P, f + g, α) ≤ U(P, f, α) + U(P, g, α) (5.19.2) and L(P, f + g, α) ≥ L(P, f, α) + L(P, g, α), (5.19.3) using the remarks about suprema and infima of sums of functions in Section 5.8.

Let P_{1} and P_{2} be arbitrary partitions of [a, b], and let P^{∗} be a common refinement of them, as in Section 5.14. Observe that

\overline{∫} _{a}^{b}(f + g) dα ≤ U(P^{∗}, f + g, α) ≤ U(P^{∗}, f, α) + U(P^{∗}, g, α)
(5.19.4)  ≤ U(P_{1}, f, α) + U(P_{2}, g, α),

where the first step is as in Section 5.13, the second step is as in (5.19.2), and the third step is as in Section 5.12. One can use this to get that

\overline{∫} _{a}^{b}(f + g) dα ≤ \overline{∫} _{a}^{b}f dα + \overline{∫} _{a}^{b}g dα. (5.19.5)

Similarly,

\underline{∫} _{a}^{b}(f + g) dα ≥ L(P^{∗}, f + g, α) ≥ L(P^{∗}, f, α) + L(P^{∗}, g, α)
(5.19.6)  ≥ L(P_{1}, f, α) + L(P_{2}, g, α).

This implies that

\underline{∫} _{a}^{b}(f + g) dα ≥ \underline{∫} _{a}^{b}f dα + \underline{∫} _{a}^{b}g dα. (5.19.7)

If f and g are Riemann–Stieltjes integrable with respect to α on [a, b], then it follows that

(5.19.8) f + g is Riemann–Stieltjes integrable with respect to α on [a, b],

with

∫_{a}^{b}(f + g) dα = ∫_{a}^{b} f dα + ∫_{a}^{b} g dα. (5.19.9)

This corresponds to another part of part (a) of Theorem 6.12 on p128 of [112].

Let f and g be any two bounded real-valued functions on [a, b] again. If P is a partition of [a, b], then one can verify that

U(P, f + g, α) ≥ U(P, f, α) + L(P, g, α) (5.19.10)

and L(P, f + g, α) ≤ L(P, f, α) + U(P, g, α), (5.19.11)

using the remarks at the beginning of Section 5.9. These inequalities can also be obtained from those mentioned at the beginning of the section using some of the remarks in the previous section.

One can use these inequalities to get that

\overline{∫} _{a}^{b}(f + g) dα ≥ \overline{∫} _{a}^{b}f dα + \underline{∫} _{a}^{b}g dα (5.19.12) and

\underline{∫} _{a}^{b}(f + g) dα ≤ \underline{∫} _{a}^{b}f dα + \overline{∫} _{a}^{b}g dα, (5.19.13)

as before. This could also be obtained from the analogous inequalities mentioned earlier and some of the remarks in the previous section. If g is Riemann–Stieltjes integrable with respect to α on [a, b], then we get that

\overline{∫} _{a}^{b}(f + g) dα = \overline{∫} _{a}^{b}f dα + ∫_{a}^{b} g dα (5.19.14)

and

\underline{∫} _{a}^{b}(f + g) dα = \underline{∫} _{a}^{b}f dα + ∫_{a}^{b} g dα. (5.19.15)

5.20 Another monotonically increasing function

We continue with the same notation and hypotheses as in the previous sections, starting in Section 5.10. Let β be another monotonically increasing real-valued function on [a, b], so that

(5.20.1) α + β is monotonically increasing on [a, b]

too. If P is a partition of [a, b], then it is easy to see that

U(P, f, α + β) = U(P, f, α) + U(P, f, β) (5.20.2)

and L(P, f, α + β) = L(P, f, α) + L(P, f, β). (5.20.3)

This implies that

U(P, f, α + β) ≥ \overline{∫} _{a}^{b}f dα + \overline{∫} _{a}^{b}f dβ (5.20.4)

and

L(P, f, α + β) ≤ \underline{∫} _{a}^{b}f dα + \underline{∫} _{a}^{b}f dβ. (5.20.5)

It follows that

\overline{∫} _{a}^{b}f d(α + β) ≥ \overline{∫} _{a}^{b}f dα + \overline{∫} _{a}^{b}f dβ (5.20.6)

and

\underline{∫} _{a}^{b}f d(α + β) ≤ \underline{∫} _{a}^{b}f dα + \underline{∫} _{a}^{b}f dβ. (5.20.7)

5.20. ANOTHER MONOTONICALLY INCREASING FUNCTION 103

Let P_{1} and P_{2} be arbitrary partitions of [a, b], and let P^{∗} be a common refinement of them. Note that

\overline{∫} _{a}^{b}f d(α + β) ≤ U(P^{∗}, f, α + β) = U(P^{∗}, f, α) + U(P^{∗}, f, β)
(5.20.8)  ≤ U(P_{1}, f, α) + U(P_{2}, f, β),

which implies that \overline{∫} _{a}^{b}f d(α + β) ≤ \overline{∫} _{a}^{b}f dα + \overline{∫} _{a}^{b}f dβ. (5.20.9) Combining this with (5.20.6) we obtain that \overline{∫} _{a}^{b}f d(α + β) = \overline{∫} _{a}^{b}f dα + \overline{∫} _{a}^{b}f dβ. (5.20.10)

Similarly,

\underline{∫} _{a}^{b}f d(α + β) ≥ L(P^{∗}, f, α + β) = L(P^{∗}, f, α)L(P^{∗}, f, β)
(5.20.11)  ≥ L(P_{1}, f, α) + L(P_{2}, f, β),

and thus \underline{∫} _{a}^{b}f d(α + β) ≥ \underline{∫} _{a}^{b}f dα + \underline{∫} _{a}^{b}f dβ. (5.20.12) This means that \underline{∫} _{a}^{b}f d(α + β) = \underline{∫} _{a}^{b}f dα + \underline{∫} _{a}^{b}f dβ, (5.20.13) because of (5.20.7).

One can use (5.20.10) and (5.20.13) to get that (5.20.14) f is Riemann–Stieltjes integrable with respect to α + β on [a, b] if and only if (5.20.15) f is Riemann–Stieltjes integrable with respect to

both α and β on [a, b].

In this case, we have that ∫_{a}^{b} f d(α + β) = ∫_{a}^{b} f dα + ∫_{a}^{b} f dβ, (5.20.16) as in part (e) of Theorem 6.12 on p128 of [112].

5.21 Some inequalities with integrals

Let us continue with the same notation and hypotheses as in the previous sections, starting in Section 5.10. Let g be another bounded real-valued function on [a, b] again, and suppose that

f(x) ≤ g(x) (5.21.1)

for every x ∈ [a, b]. If P is a partition of [a, b], then it is easy to see that

U(P, f, α) ≤ U(P, g, α) (5.21.2)

and L(P, f, α) ≤ L(P, g, α). (5.21.3)

This implies that

\overline{∫} _{a}^{b}f dα ≤ U(P, g, α) (5.21.4)

and

L(P, f, α) ≤ \underline{∫} _{a}^{b}g dα, (5.21.5)

by definition of the upper and lower integrals, as in Section 5.13. It follows that

\overline{∫} _{a}^{b}f dα ≤ \overline{∫} _{a}^{b}g dα (5.21.6)

and

\underline{∫} _{a}^{b}f dα ≤ \underline{∫} _{a}^{b}g dα, (5.21.7)

because P is arbitrary. If f and g are Riemann–Stieltjes integrable with respect to α on [a, b], then we get that

∫_{a}^{b} f dα ≤ ∫_{a}^{b} g dα. (5.21.8)

This corresponds to part (b) of Theorem 6.12 on p128 of [112].

If f is Riemann–Stieltjes integrable with respect to α on [a, b], then

( inf_{a≤x≤b} f(x)) (α(b) − α(a)) ≤ ∫_{a}^{b} f dα ≤ ( sup_{a≤x≤b} f(x)) (α(b) − α(a)) (5.21.9)

as in Section 5.13. The first inequality implies that

(5.21.10)  − ∫_{a}^{b} f dα ≤ −( inf_{a≤x≤b} f(x)) (α(b) − α(a))
= ( sup_{a≤x≤b}(−f(x))) (α(b) − α(a)),

5.22. SUBINTERVALS OF [A, B] 105

where the second step is as in Section 5.8. One can check that

(5.21.11)  sup_{a≤x≤b} |f(x)| = max ( sup_{a≤x≤b} f(x), sup_{a≤x≤b}(−f(x))).

It follows that

|∫_{a}^{b} f dα| ≤ ( sup_{a≤x≤b} |f(x)|) (α(b) − α(a)). (5.21.12)

This corresponds to part (d) of Theorem 6.12 on p128 of [112].

5.22 Subintervals of [a, b]

We continue with the same notation and hypotheses as in the previous sections again, starting in Section 5.10. Let c be a real number with

a ≤ c ≤ b, (5.22.1)

so that [a, c] and [c, b] are closed intervals in R with

[a, c] ∪ [c, b] = [a, b]. (5.22.2)

If P_{1} and P_{2} are partitions of [a, c] and [c, b], respectively, then we can combine them to get a partition P_{1,2} of [a, b]. Of course, the restrictions of f to [a, c] and [c, b] define bounded real-valued functions on the two intervals. Similarly, the restrictions of α to [a, c] and [c, b] define monotonically increasing real-valued functions on these intervals. It is easy to see that

U(P_{1,2}, f, α) = U(P_{1}, f, α) + U(P_{2}, f, α) (5.22.3)

and L(P_{1,2}, f, α) = L(P_{1}, f, α) + L(P_{2}, f, α). (5.22.4)

This implies that

\overline{∫} _{a}^{b}f dα ≤ U(P_{1}, f, α) + U(P_{2}, f, α) (5.22.5)

and

\underline{∫} _{a}^{b}f dα ≥ L(P_{1}, f, α) + L(P_{2}, f, α). (5.22.6)

One can use this to get that

\overline{∫} _{a}^{b}f dα ≤ \overline{∫} _{a}^{c}f dα + \overline{∫} _{c}^{b}f dα (5.22.7)

and

\underline{∫} _{a}^{b}f dα ≥ \underline{∫} _{a}^{c}f dα + \underline{∫} _{c}^{b}f dα. (5.22.8)

Let P be any partition of [a, b], and let P^{∗} be a partition of [a, b] that is a refinement of P that contains c, which can be obtained by adding c to P, if necessary. Using P^{∗}, we get partitions P_{1}^{∗} and P_{2}^{∗} of [a, c] and [c, b], respectively. Observe that

(5.22.9) U(P, f, α) ≥ U(P^{∗}, f, α) = U(P_{1}^{∗}, f, α) + U(P_{2}^{∗}, f, α)
≥ \overline{∫} _{a}^{c}f dα + \overline{∫} _{c}^{b}f dα,

where the first step is as in Section 5.12, the second step is as in (5.22.3), and the third step is as in Section 5.13. This implies that

\overline{∫} _{a}^{b}f dα ≥ \overline{∫} _{a}^{c}f dα + \overline{∫} _{c}^{b}f dα, (5.22.10)

so that

\overline{∫} _{a}^{b}f dα = \overline{∫} _{a}^{c}f dα + \overline{∫} _{c}^{b}f dα. (5.22.11)

Similarly,

(5.22.12) L(P, f, α) ≤ L(P^{∗}, f, α) = L(P_{1}^{∗}, f, α) + L(P_{2}^{∗}, f, α)
≤ \underline{∫} _{a}^{c}f dα + \underline{∫} _{c}^{b}f dα,

which implies that

\underline{∫} _{a}^{b}f dα ≤ \underline{∫} _{a}^{c}f dα + \underline{∫} _{c}^{b}f dα, (5.22.13)

and thus

\underline{∫} _{a}^{b}f dα = \underline{∫} _{a}^{c}f dα + \underline{∫} _{c}^{b}f dα. (5.22.14)

These statements could also be obtained from the previous ones using the remarks in Section 5.18.

One can use (5.22.11) and (5.22.14) to get that f is Riemann–Stieltjes integrable with respect to α on [a, b] if and only if

(5.22.15) the restrictions of f to [a, c] and [c, b] are Riemann–Stieltjes integrable with respect to α.

In this case, we have that

∫_{a}^{b} f dα = ∫_{a}^{c} f dα + ∫_{c}^{b} f dα, (5.22.16)

as in part (c) of Theorem 6.12 on p128 of [112].

If f is Riemann–Stieltjes integrable with respect to α on [a, b], then one can repeat the argument in the preceding paragraph to get that

(5.22.17) the restriction of f to any closed subinterval of [a, b] is Riemann–Stieltjes integrable with respect to α

as well.

5.23 Derivatives of integrals

We continue with the same notation and hypotheses as before, starting in Section 5.10. Suppose that f is Riemann–Stieltjes integrable with respect to α on [a, b]. If x ∈ [a, b], then the restriction of f to [a, x] is Riemann–Stieltjes integrable with respect to α, as in the previous section, and we put

F (x) = ∫_{a}^{x} f dα. (5.23.1)

Similarly, if y ∈ [x, b], then the restrictions of f to [a, y] and [x, y] are Riemann– Stieltjes integrable with respect to α, and

F (y) − F (x) = ∫_{x}^{y} f dα, (5.23.2)

as in the previous section.

Observe that

|F (y) − F (x)| = |∫_{x}^{y} f dα| ≤ ( sup_{x≤v≤y} |f(v)|) (α(y) − α(x))
(5.23.3)  ≤ ( sup_{a≤v≤b} |f(v)|) (α(y) − α(x)),

where the second step is as in Section 5.21. One can use this to get that

(5.23.4) F is continuous at any point in [a, b] at which α is continuous.

This basically corresponds to the first part of Theorem 6.20 on p133 of [112].

Suppose from now on in this section that a < b, and that α(u) = u on [a, b], so that f is Riemann integrable on [a, b]. Put

F (x) = ∫_{a}^{x} f(t) dt (5.23.5)

for each x ∈ [a, b], as before. If y ∈ [x, b], then

F (y) − F (x) = ∫_{x}^{y} f(t) dt, (5.23.6) as before. If (5.23.7) f is continuous at a point w ∈ [a, b],

then it is well known and not too difficult to show that

(5.23.8) F is differentiable at w,

with F′(w) = f(w). (5.23.9)

This is the second part of Theorem 6.20 on p133 of [112].

More precisely, suppose first that

a ≤ w < b, (5.23.10)

and let h > 0 be given, with h ≤ b − w. Observe that

\frac{F (w + h) − F (w)}{h} = \frac{1}{h} ∫_{w}^{w+h} f(t) dt, (5.23.11)

because of (5.23.6). If f is continuous at w, then one can check that

lim_{h→0+} \frac{F (w + h) − F (x)}{h} = f(w). (5.23.12)

Of course, if f is constant on [a, b], then (5.23.11) is the same as the constant value of f. One can use this to reduce to the case where f(w) = 0, which is a bit simpler.

Similarly, suppose now that

a < w ≤ b, (5.23.13)

and let h > 0 be given, with h ≤ w − a. In this case,

\frac{F (w) − F (w − h)}{h} = \frac{1}{h} ∫_{w−h}^{w} f(t) dt, (5.23.14)

using (5.23.6) again. If f is continuous at w, then one can verify that

lim_{h→0+} \frac{F (w) − F (w − h)}{h} = f(w), (5.23.15)

as before.

5.24 Integrals of derivatives

Let a and b be real numbers with a < b, and let F_{1} be a real-valued function on [a, b]. Suppose that

(5.24.1) F_{1} is differentiable at every point in [a, b], and put f = F′_{1} (5.24.2) on [a, b]. If (5.24.3) f is Riemann integrable on [a, b], then Theorem 6.21 on p134 of [112] says that

∫_{a}^{b} f(x) dx = F_{1}(b) − F_{1}(a). (5.24.4)

This is a version of the fundamental theorem of calculus. Of course, (5.24.3) includes the condition that f be bounded on [a, b].

If f is continuous on [a, b], and F is as in (5.23.5), then F is differentiable at every point in [a, b], as in the previous section, and we have that

(F_{1} − F)′ = F′_{1} − F′ = 0 (5.24.5)

on [a, b]. This means that F_{1}−F is constant on [a, b], by the mean value theorem. In this case, (5.24.4) follows from the definition of F.

Otherwise, let P = {t_{j}}_{j=0}^{l} be a partition of [a, b], and note that

∑_{j=1}^{l}(F_{1}(t_{j}) − F_{1}(t_{j−1})) = F_{1}(b) − F_{1}(a), (5.24.6)

as in (5.11.5). The mean value theorem implies that for each j = 1, …, l there is an r_{j} ∈ [t_{j−1}, t_{j}] such that

F_{1}(t_{j}) − F_{1}(t_{j−1}) = F′_{1}(r_{j}) (t_{j} − t_{j−1}). (5.24.7)

This means that

∑_{j=1}^{l} F′_{1}(r_{j}) (t_{j} − t_{j−1}) = F_{1}(b) − F_{1}(a), (5.24.8)

by (5.24.6).

If f is bounded on [a, b], then

L(P, f) ≤ ∑_{j=1}^{l} F′_{1}(r_{j}) (t_{j} − t_{j−1}) ≤ U(P, f), (5.24.9)

by definition of the upper and lower Darboux sums associated to f and P, as in Section 5.10. Thus

L(P, f) ≤ F_{1}(b) − F_{1}(a) ≤ U(P, f), (5.24.10)

by (5.24.8). It follows that

\underline{∫} _{a}^{b}f(x) dx ≤ F_{1}(b) − F_{1}(a) ≤ \overline{∫} _{a}^{b}f(x) dx, (5.24.11)

by definition of the upper and lower integrals, as in Section 5.13. If (5.24.3) holds, then we get that (5.24.4) holds.

110 CHAPTER 5. SOME DERIVATIVES AND INTEGRALS

5.25 More on Riemann–Stieltjes integrability

Let us continue with the same notation and hypotheses as before, starting in Section 5.10. Let a_{1}, b_{1} be real numbers with a_{1} ≤ b_{1}, and suppose that a_{1} ≤ f ≤ b_{1} (5.25.1) on [a, b]. Also let ϕ be a continuous real-valued function on [a_{1}, b_{1}], and note that ϕ is bounded on [a_{1}, b_{1}], because [a_{1}, b_{1}] is compact. Thus ϕ∘f is a bounded real-valued function on [a, b].

If f is Riemann–Stieltjes integrable with respect to α on [a, b], then (5.25.2) ϕ ∘ f is Riemann–Stieltjes integrable with respect to α on [a, b] too. This is Theorem 6.11 on p127 of [112]. This uses the fact that ϕ is uniformly continuous on [a_{1}, b_{1}], because [a_{1}, b_{1}] is compact. Basically, one can show that U(P, ϕ ∘ f, α) − L(P, ϕ ∘ f, α) (5.25.3) is arbitrarily small when U(P, f, α) − L(P, f, α) (5.25.4) is sufficiently small.

In particular, if f is a bounded real-valued function on [a, b] that is Riemann– Stieltjes integrable with respect to α, then (5.25.5) f^{2} is Riemann–Stieltjes integrable with respect to α on [a, b]. Let g be another bounded real-valued function on [a, b] that is Riemann–Stieltjes integrable with respect to α. Under these conditions, part (a) of Theorem 6.13 on p129 of [112] says that (5.25.6) f g is Riemann–Stieltjes integrable with respect to α on [a, b]. This uses the fact that f g = (1/2) ((f + g)^{2} − f^{2} − g^{2}). (5.25.7)

Similarly, (5.25.8) |f| is Riemann–Stieltjes integrable with respect to α on [a, b], as in part (b) of Theorem 6.13 on p129 of [112]. Note that

∫_{a}^{b} f dα, − ∫_{a}^{b} f dα ≤ ∫_{a}^{b} |f| dα, (5.25.9) because f, −f ≤ |f| on [a, b]. This means that

|∫_{a}^{b} f dα| ≤ ∫_{a}^{b} |f| dα, (5.25.10)

as in [112].

5.26. DIFFERENTIABILITY OF α 111

5.26 Differentiability of α

Let us continue with the same notation and hypotheses as in the previous sections. Suppose that a < b, (5.26.1) α is differentiable at every point in [a, b], α′ is bounded on [a, b], and (5.26.2) α′ is Riemann integrable on [a, b]. Under these conditions, Theorem 6.17 on 131 of [112] states that a bounded real-valued function f on [a, b] is Riemann–Stieltjes integrable with respect to α on [a, b] if and only if (5.26.3) f α′ is Riemann integrable on [a, b], in which case ∫_{a}^{b} f dα = ∫_{a}^{b} f(x) α′(x) dx. (5.26.4) Note that (5.26.3) holds when f is Riemann integrable on [a, b], because of (5.26.2), as in the previous section.

5.27 Changing variables

Let a_{0} and b_{0} be real numbers with a_{0} ≤ b_{0}, and let ϕ be a continuous realvalued function on [a_{0}, b_{0}] that is also monotonically increasing on [a_{0}, b_{0}]. If y, z are real numbers with a_{0} ≤ y ≤ z ≤ b_{0}, then ϕ([y, z]) = [ϕ(y), ϕ(z)]. (5.27.1) More precisely, ϕ([y, z]) ⊆ [ϕ(y), ϕ(z)], (5.27.2) because ϕ is monotonically increasing on [a_{0}, b_{0}], and [ϕ(y), ϕ(z)] ⊆ ϕ([y, z]), (5.27.3) by the intermediate value theorem.

Put a = ϕ(a_{0}) and b = ϕ(b_{0}), so that a ≤ b. Let α be a monotonically increasing real-valued function on [a, b], as usual, and put β = α ∘ ϕ. (5.27.4) It is easy to see that this is a monotonically increasing real-valued function on [a_{0}, b_{0}]. Also let f be a bounded real-valued function on [a, b], as before, so that g = f ∘ ϕ (5.27.5) is a bounded real-valued function on [a_{0}, b_{0}].

Let P_{0} = {t_{0,j}}_{j=0}^{l_{0}} be a partition of [a_{0}, b_{0}]. Note that

P_{0}^{ϕ} = {ϕ(t_{0,j})}_{j=0}^{l_{0}} (5.27.6)

is a partition of [a, b]. One can check that

U(P_{0}, g, β) = U(P_{0}^{ϕ}, f, α) (5.27.7)

and L(P_{0}, g, β) = L(P_{0}^{ϕ}, f, α), (5.27.8)

as in (38) on p133 of [112]. This uses the fact that

ϕ([t_{0,j−1}, t_{0,j}]) = [ϕ(t_{0,j−1}), ϕ(t_{0,j})] (5.27.9)

for each j = 1, …, l_{0}, as in (5.27.1).

Observe that every partition of [a, b] is of the form (5.27.6) for some partition P_{0} of [a_{0}, b_{0}]. One can use this and (5.27.7), (5.27.8) to get that

\overline{∫} _{a_{0}}^{b_{0}}g dβ = \overline{∫} _{a}^{b}f dα (5.27.10)

and

\underline{∫} _{a_{0}}^{b_{0}}g dβ = \underline{∫} _{a}^{b}f dα. (5.27.11)

In particular, if f is Riemann–Stieltjes integrable with respect to α on [a, b], then

(5.27.12) g is Riemann–Stieltjes integrable with respect to β on [a_{0}, b_{0}],

with

∫_{a_{0}}^{b_{0}} g dβ = ∫_{a}^{b} f dα, (5.27.13)

as in Theorem 6.19 on p132 of [112]. Although this theorem is stated for the case where ϕ is strictly increasing on [a_{0}, b_{0}], we do not really need this here. If ϕ is strictly increasing on [a_{0}, b_{0}], then we get a one-to-one correspondence between partitions of [a_{0}, b_{0}] and [a, b], but this is not really needed.

Of course, if ϕ(y) = ϕ(z) (5.27.14)

for some y, z ∈ [a_{0}, b_{0}] with y < z, then ϕ is constant on [y, z], because ϕ is monotonically increasing on [a_{0}, b_{0}]. This implies that g and β are constant on [y, z] as well.

More precisely, the previous argument shows that (5.27.12) holds if and only if f is Riemann–Stieltjes integrable with respect to α on [a, b]. If ϕ is strictly increasing on [a_{0}, b_{0}], then one can also look at the “only if” part in terms of the inverse of ϕ.

5.28 Some sums of step functions

We continue with the same notation and hypotheses as before, starting in Section 5.10 again. Let I(x) be the unit step function on the real line, as in Section 4.12, and let {w_{j}}_{j=1}^{∞} be a sequence of elements of (a, b). Also let {c_{j}}_{j=1}^{∞} be an infinite sequence of nonnegative real numbers such that

(5.28.1)  ∑_{j=1}^{∞} c_{j} converges,

and put

α(x) = ∑_{j=1}^{∞} c_{j} I(x − w_{j}) (5.28.2)

for each x ∈ [a, b], where the series on the right converges by the comparison test. This defines a monotonically increasing real-valued function on [a, b]. If f is continuous on [a, b], then Theorem 6.16 on p130 of [112] states that

∫_{a}^{b} f dα = ∑_{j=1}^{∞} c_{j} f(w_{j}). (5.28.3)

Note that the series on the right converges absolutely, by the comparison test, because f is bounded on [a, b]. More precisely, in [112], the w_{j}’s are asked to be distinct, but this does not really seem to be needed, although there may not be any reason to consider repetitions. One could permit w_{j} to be equal to a for some j, but not b, because of the way that I(x) is defined, as in Section 5.17. One could use analogous arguments to get that f is Riemann–Stieltjes integrable with respect to α on [a, b] when

(5.28.4) f is continuous at w_{j} for each j,

in addition to being bounded on [a, b], and that (5.28.3) holds in this case. Let n be a positive integer, and put

α_{n}(x) = ∑_{j=1}^{n} c_{j} I(x − w_{j}) (5.28.5)

for each x ∈ [a, b]. This is a monotonically increasing real-valued function on [a, b]. If f is continuous at each of w_{1}, …, w_{n}, then it is easy to see that f is Riemann–Stieltjes integrable with respect to α_{n} on [a, b], with

∫_{a}^{b} f dα_{n} = ∑_{j=1}^{n} c_{j} f(w_{j}), (5.28.6)

using remarks in Section 5.17 and 5.20.

Similarly,

β_{n}(x) = ∑_{j=n+1}^{∞} c_{j} I(x − w_{j}) (5.28.7)

defines a monotonically increasing real-valued function on [a, b], with

α_{n}(x) + β_{n}(x) = α(x) (5.28.8)

for each x ∈ [a, b]. Observe that

\overline{∫} _{a}^{b}f dβ_{n} ≤ M (β_{n}(b) − β_{n}(a)) = M ∑_{j=n+1}^{∞} c_{j} (5.28.9)

and

\underline{∫} _{a}^{b}f dβ_{n} ≥ m (β_{n}(b) − β_{n}(a)) = m ∑_{j=n+1}^{∞} c_{j}, (5.28.10)

where the first steps are as in Section 5.13, and the second steps follow from the definition (5.28.7) of β_{n}. Of course, the right sides of (5.28.9) and (5.28.10) tend to 0 as n → ∞, because ∑_{j=1}^{∞} c_{j} converges, by hypothesis. To get the conclusions mentioned earlier, one can use (5.28.8) and the remarks in Section 5.20.

5.29 Limits of integrals

Let a and b be real numbers with a < b, and let {f_{j}}_{j=1}^{∞} be a sequence of bounded real-valued functions on [a, b] that converges uniformly to a real-valued function f on [a, b]. Note that (5.29.1) f is bounded on [a, b]

as well, as mentioned in Section 4.9. Let α be a monotonically increasing realvalued function on [a, b], as before.

Of course, {f_{j} − f}_{j=1}^{∞} converges to 0 uniformly on [a, b]. It is easy to see that

lim_{j→∞} \overline{∫} _{a}^{b}(f_{j} − f) dα = 0 (5.29.2)

and

lim_{j→∞} \underline{∫} _{a}^{b}(f_{j} − f) dα = 0, (5.29.3)

using the remarks in Section 5.13.

It is not too difficult to show that

lim_{j→∞} \overline{∫} _{a}^{b}f_{j} dα = \overline{∫} _{a}^{b}f dα (5.29.4)

and

lim_{j→∞} \underline{∫} _{a}^{b}f_{j} dα = \underline{∫} _{a}^{b}f dα, (5.29.5)

using the remarks in Section 5.19. This also uses (5.29.2), (5.29.3), the facts that f_{j} = f + (f_{j} − f) (5.29.6)

and f = f_{j} + (f − f_{j}) (5.29.7)

for each j.

If

(5.29.8) f_{j} is Riemann–Stieltjes integrable with respect to α on [a, b]

for each j, then it follows that

(5.29.9) f is Riemann–Stieltjes integrable with respect to α on [a, b],

with

lim_{j→∞} ∫_{a}^{b} f_{j} dα = ∫_{a}^{b} f dα, (5.29.10)

as in Theorem 7.16 on p151 of [112].

Bibliography

[1] R. Agarwal, C. Flaut, and D. O’Regan, An Introduction to Real Analysis, CRC Press, 2018.

[2] T. Apostol, Mathematical Analysis, 2nd edition, Addison-Wesley, 1974.

[3] R. Beals, Analysis: An Introduction, Cambridge University Press, 2004.

[4] R. Beals, The scope of the power series method, Mathematics Magazine 86 (2013), 56–62.

[5] C. Belna, M. Evans, and P. Humke, Symmetric and strong differentiation, American Mathematical Monthly 86 (1979), 121–123.

[6] L. Bers, Classroom notes: On avoiding the mean value theorem, American Mathematical Monthly 74 (1967), 583.

[7] R. Boas, A Primer of Real Functions, 4th edition, revised and updated by H. Boas, Mathematical Association of America, 1996.

[8] S. Bochner, Fourier series came first, American Mathematical Monthly 86 (1979), 197–199.

[9] M. Botsko, An easy generalization of the Riemann integral, American Mathematical Monthly 93 (1986), 728–732.

[10] M. Botsko, An elementary proof that a bounded a.e. continuous function is Riemann integrable, American Mathematical Monthly 95 (1988), 249– 252.

[11] M. Botsko, A fundamental theorem of calculus that applies to all Riemann integrable functions, Mathematics Magazine 64 (1991), 347–348.

[12] M. Botsko, Exactly which bounded Darboux functions are derivatives?, American Mathematical Monthly 114 (2007), 242–246.

[13] D. Bressoud, A Radical Approach to Real Analysis, 2nd edition, Mathematical Association of America, 2007.

[14] D. Bressoud, A Radical Approach to Lebesgue’s Theory of Integration, MAA Textbooks, Cambridge University Press, 2008.

[15] D. Bressoud, Historical reflections on teaching the fundamental theorem of integral calculus, American Mthematical Monthly 118 (2011), 99–115.

[16] D. Bressoud, Calculus Reordered: A History of the Big Ideas, Princeton University Press, 2019.

[17] A. Browder, Mathematical Analysis: An Introduction, Springer-Verlag, 1996.

[18] A. Brown, A proof of the Lebesgue condition for Riemann integrability, American Mathematical Monthly 43 (1936), 396–398.

[19] A. Bruckner, Derivatives: why they elude classification, Mathematics Magazine 49 (1976), 5–11.

[20] A. Bruckner, Differentiation of Real Functions, 2nd edition, American Mathematical Society, 1994.

[21] A. Bruckner, J. Mařík, and C. Weil, Some aspects of products of derivatives, American Mathematical Monthly 99 (1992), 134–145.

[22] G. Bullock, The teaching of mathematics: A geometric interpretation of the Riemann–Stieltjes integral, American Mathematical Monthly 95 (1988), 448–455.

[23] J. Burkill and H. Burkill, A Second Course in Mathematical Analysis, Cambridge University Press, 1970.

[24] L. Cohen, Classroom notes: On being mean to the mean value theorem, American Mathematical Monthly 74 (1967), 581–582.

[25] J. Conway, A First Course in Analysis, Cambridge University Press, 2018.

[26] A. Devinatz, Advanced Calculus, Holt, Rinehart and Winston, 1968.

[27] T. Drammatico, A new proof of the equality of mixed second partial derivatives, American Mathematical Monthly 122 (2015), 602–603.

[28] R. Dressler and K. Stromberg, The Tonelli integral, American Mathematical Monthly 81 (1974), 67–68.

[29] W. Dunham, Touring the calculus gallery, American Mathematical Monthly 112 (2005), 1–19.

[30] W. Dunham, The Calculus Gallery: Masterpieces from Newton to Lebesgue, Princeton University Press, 2018.

[31] M. Evans and P. Humke, Revisiting a century-old characterization of Baire class one, Darboux functions, American Mathematical Monthly 116 (2009), 451-455.

118 BIBLIOGRAPHY

[32] W. Felscher, Bolzano, Cauchy, epsilon, delta, American Mathematical Monthly 107 (2000), 844–862.

[33] L. Flatto, Advanced Calculus, Williams & Wilkins, 1976.

[34] M. Fort, Jr., Classroom notes: The maximum value of a continuous function, American Mathematical Monthly 58 (1951), 32–33.

[35] M. Fort, Jr., Mathematical notes: A theorem concerning functions discontinuous on a dense set, American Mathemtical Monthly 58 (1951), 408–410.

[36] M. Fort, Jr., Classroom notes: A fundamental theorem of calculus, American Mathematical Monthly 63 (1956), 334–335.

[37] P. Franklin, A Treatise on Advanced Calculus, Dover, 1964.

[38] J. Franks, Cantor’s other proofs that R is uncountable, Mathematics Magazine 83 (2010), 283–289.

[39] A. Friedman, Advanced Calculus, Holt, Rinehart and Winston, 1971.

[40] H. Gaskill and P. Narayanaswami, Foundations of Analysis: The Theory of Limits, Harper & Row, 1989.

[41] B. Gelbaum and J. Olmstead, Counterexamples in Analysis, 2nd edition, Dover, 2003.

[42] K. Gödel, What is Cantor’s continuum problem?, American Mathematical Monthly 54 (1947), 515–525.

[43] C. Goffman, Definition of the Lebesgue integral, American Mathematical Monthly 60 (1953), 251–252.

[44] C. Goffman, Classroom notes: Arc length, American Mathematical Monthly 71 (1964), 303–304.

[45] C. Goffman, Introduction to Real Analysis, Harper & Row, 1966.

[46] C. Goffman, A bounded derivative which is not Riemann integrable, American Mathematical Monthly 84 (1977), 205–206.

[47] R. Goldberg, Methods of Real Analysis, 2nd edition, Wiley, 1976.

[48] E. González-Velasco, Connections in mathematical analysis: the case of Fourier series, American Mathematical Monthly 99 (1992), 427–441.

[49] E. González-Velasco, James Gregory’s calculus in the Geometriæ pars universalis, American Mathematical Monthly 114 (2007), 565–576.

[50] E. González-Velasco, Journey Through Mathematics: Creative Episodes in its History, Springer, 2011.

[51] R. Gordon, A bounded derivative that is not Riemann integrable, Mathematics Magazine 89 (2016), 364–370.

[52] F. Gouvêa, Was Cantor surprised?, American Mathematical Monthly 118 (2011), 198–209.

[53] J. Grabiner, The Origins of Cauchy’s Rigorous Calculus, MIT Press, 1981.

[54] J. Grabiner, Who gave you the epsilon? Cauchy and the origins of rigorous calculus, American Mathematical Monthly 90 (1983), 185–194.

[55] J. Grabiner, The changing concept of change: the derivative from Fermat to Weierstrass, Mathematics Magazine 56 (1983), 195–206.

[56] J. Grabiner, Was Newton’s calculus a dead end? The continental influence of Maclaurin’s treatise of fluxions, American Mathematical Monthly 104 (1997), 393–410.

[57] J. Grabiner, A Historian Looks Back: The Calculus as Algebra and Selected Writings, Mathematical Association of America, 2010.

[58] J. Gray, Miscellanea: Clear theory, American Mathematical Monthly 92 (1985), 158–159.

[59] J. Gray, The Real and the Complex: A History of Analysis in the 19th Century, Springer, 2015.

[60] J. Green, The theory of functions of a real variable, Mathematics Magazine 24 (1951), 209–217.

[61] R. Grinberg, The Real Analysis Lifesaver: All the Tools that You Need to Understand Proofs, Princeton University Press, 2017.

[62] R. Gunning, An Introduction to Analysis, Princeton University Press, 2018.

[63] H. Herrlich, Axiom of Choice, Lecture Notes in Mathematics 1876, Springer-Verlag, 2006.

[64] T. Hildebrandt, Definitions of Stieltjes integrals of the Riemann type, American Mathematical Monthly 45 (1938), 265–278.

[65] K. Hoffman, Analysis in Euclidean Space, Prentice-Hall, 1975.

[66] F. Jones, Lebesgue Integration on Euclidean Space, Jones and Bartlett, 1993.

[67] H. Junghenn, A Course in Real Analysis, CRC Press, 2015.

[68] W. Kaczor and M. Nowak, Problems in Mathematical Analysis I: Real Numbers, Sequences and Series, translated and revised from the 1996 Polish original by the authors, American Mathematical Society, 2000.

120 BIBLIOGRAPHY

[69] W. Kaczor and M. Nowak, Problems in Mathematical Analysis II: Continuity and Differentiation, translated from the 1998 Polish original, revised, and augmented by the authors, American Mathematical Society, 2001.

[70] W. Kaczor and M. Nowak, Problems in Mathematical Analysis III: Integration, American Mathematical Society, 2003.

[71] I. Kaplansky, Set Theory and Metric Spaces, 2nd edition, Chelsea, 1977.

[72] H. Kaptanoğlu, In praise of y = x^{α} sin(\frac{1}{x}), American Mathematical Monthly 108 (2001), 144–150.

[73] G. Klambauer, Mathematical Analysis, Dekker, 1975.

[74] A. Knapp, Basic Real Analysis, Birkhäuser, 2005.

[75] J. Koliha, Mean, meaner, and the meanest mean value theorem, American Mathematical Monthly 116 (2009), 356–361.

[76] T. Körner, Differentiable functions on the rationals, Bulletin of the London Mathematical Society 23 (1991), 557–562.

[77] T. Körner, Butterfly hunting and the symmetry of mixed partial derivativs, American Mathematical Monthly 105 (1998), 756–758.

[78] T. Körner, A Companion to Analysis: A Second First and First Second Course in Analysis, American Mathematical Society, 2004.

[79] S. Krantz, A Guide to Real Variables, Mathematical Association of America, 2009.

[80] S. Krantz, Foundations of Analysis, CRC Press, 2015.

[81] S. Krantz, Real Analysis and Foundations, 4th edition, CRC Press, 2017.

[82] S. Krantz, The Elements of Advanced Mathematics, 4th edition, CRC Press, 2018.

[83] S. Krantz, Elementary Introduction to the Lebesgue Integral, CRC Press, 2018.

[84] M. Laczkovich and V. Sós, Real Analysis — Foundations and Functions of One Variable, translated from the third (2005) Hungarian edition by the authors, Springer, 2015.

[85] S. Lang, Undergraduate Analysis, 2nd edition, Springer-Verlag, 1997.

[86] R. Langer, Fourier’s Series: The Genesis and Evolution of a Theory, The first Herbert Ellsworth Slaught Memorial Paper, American Mathematical Monthly 54 (1947), no. 7, part II.

BIBLIOGRAPHY 121

[87] W. Langlois and L. Holder, The relation of f′_{+}(a) to f′(a+), Mathematics Magazine 39 (1966), 112–120.

[88] J. LaVita, Classroom notes: A necessary and sufficient condition for Riemann integration, American Mathematical Monthly 71 (1964), 193–196.

[89] P. Lax, Rethinking the Lebesgue integral, American Mathematical Monthly 116 (2009), 863–881.

[90] T. Lehrer, The Derivative Song, American Mathematical Monthly 81 (1974), 490.

[91] T. Lehrer, There’s a Delta for Every Epsilon (Calypso), American Mathematical Monthly 81 (1974), 612.

[92] T. Lehrer, The Professor’s Song, American Mathematical Monthly 81 (1974), 745.

[93] L. Levine, On a necessary and sufficient condition for Riemann integrability, American Mathematical Monthly 84 (1977), 205.

[94] S. Malik and S. Arora, Mathematical Analysis, 2nd edition, Wiley, 1992.

[95] R. Metzler, On Riemann integrability, American Mathematical Monthly 78 (1971), 1129–1131.

[96] G. Moore, Zermelo’s Axiom of Choice: Its Origins, Development, and Influence, Springer-Verlag, 1982.

[97] F. Morgan, Real Analysis, American Mathematical Society, 2005.

[98] F. Morgan, Real Analysis and Applications: Including Fourier Series and the Calculus of Variations, American Mathematical Society, 2005.

[99] S. Nadler, Jr., A Proof of Darboux’s theorem, American Mathematical Monthly 117 (2010), 174–175.

[100] L. Olsen, A new proof of Darboux’s theorem, American Mathematical Monthly 111 (2004), 713–715.

[101] S. Ponnusamy, Foundations of Mathematical Analysis, Birkhäuser / Springer, 2012.

[102] M. Protter, Basic Elements of Real Analysis, Springer-Verlag, 1998.

[103] M. Protter and C. Morrey, Jr., A First Course in Analysis, 2nd edition, Springer-Verlag, 1991.

[104] C. Pugh, Real Mathematical Analysis, 2nd edition, Springer, 2015.

[105] H. Rademacher, On the condition of Riemann integrability, American Mathematical Monthly 61 (1954), 1–8.

[106] M. Reed, Fundamental Ideas of Analysis, Wiley, 1998.

[107] M. Rosenlicht, Introduction to Analysis, Dover, 1986.

[108] C. Rosentrater, Varieties of Integration, Mathematical Association of America, 2015.

[109] K. Ross, Another approach to Riemann–Stieltjes integrals, American Mathematical Monthly 87 (1980), 660–662.

[110] K. Ross, Elementary Analysis: The Theory of Calculus, 2nd edition, in collaboration with J. López, Springer, 2013.

[111] W. Rudin, Real and Complex Analysis, 2nd edition, McGraw-Hill, 1974. A third edition edition was published in 1987.

[112] W. Rudin, Principles of Mathematical Analysis, 3rd edition, McGraw-Hill, 1976.

[113] U. Satyanarayana, A note on Riemann-Stieltjes integrals, American Mathematical Monthly 87 (1980), 477–478.

[114] D. Schattschneider, A multiplicative metric, Mathematics Magazine 49 (1976), 203–2-5.

[115] R. Shakarchi, Problems and Solutions for Undergraduate Analysis ([85]), Springer-Verlag, 1998.

[116] O. Shisha, An approach to Darboux-Stieltjes integration, American Mathematical Monthly 72 (1965), 890–892.

[117] R. Strichartz, The Way of Analysis, Revised edition, Jones and Bartlett, 2000.

[118] K. Stromberg, An Introduction to Classical Real Analysis, AMS Chelsea, 1981.

[119] A. Taylor, Derivatives in the calculus, American Mathematical Monthly 49 (1942), 631–642.

[120] A. Taylor, L’Hospital’s rule, American Mathematical Monthly 59 (1952), 20–24.

[121] A. Taylor and W. Mann, Advanced Calculus, 3rd edition, Wiley, 1983.

[122] H. ter Horst, Riemann–Stieltjes and Lebesgue–Stieltjes integrability, American Mathematical Monthly 91 (1984), 551–559.

[123] T. Tokieda, A mean value theorem, American Mathematical Monthly 106 (1999), 673–674.

[124] D. Widder, Advanced Calculus, 3rd edition, Dover, 1989.

[125] B. Youse, Introduction to Real Analysis, Allyn and Bacon, 1972.

Index

(a, b), 5 [a, b], 6

absolute convergence, 45 absolute value, 1, 34 alternating series test, 50 archimedean property, 3

B(x, r), 4 Baire category theorem, 78 Baire class one, 86 bases, 26 bounded functions, 70 bounded sequences, 36 bounded sets, 10 Brouwer’s fixed-point theorem, 78

C, 34 C(M, N), 70 C_{b}(M, N), 70 Cantor set, 32 Cartesian products, 18 Cauchy products, 58 Cauchy sequences, 40 Cauchy’s condensation test, 48 cells, 18 chain rule, 82 closed intervals, 6 closed sets, 6 closures, 8 compact sets, 9 comparison test, 46 complements, 6 complete metric spaces, 42 completions, 78 complex conjugate, 34 compositions of mappings, 62 connected sets, 31 continuity at a point, 60 continuous mappings, 61 contraction mapping theorem, 78 contractions, 78 convergent sequences, 35 convergent series, 44 convex sets, 77 countably infinite sets, 22

Darboux sums, 91 Darboux sums, 92 Darboux–Stieltjes sums, 92 dense sets, 25 derivative, 80

from the left, 84 from the right, 83 diagonalization argument, 24 diameter, 18 differentiable functions, 80 discrete metric, 2

Euclidean metric, 1, 2 extended real numbers, 50 extreme value theorem, 64

fundamental theorem of calculus, 109

generalized mean value theorem, 85 geometric series, 46 greatest lower bounds, 3, 50

imaginary part, 34 infimum, 3, 50 intermediate value theorem, 67 intervals, 5, 6 inverse images, 61 isometries, 78 least upper bound property, 3 least upper bounds, 2, 50 Lebesgue integrals, 78 limit points, 5 limit inferior, 53 limit of a function, 72 limit point property, 10 limit superior, 53 limits of sequences, 35 Lindelöf’s theorem, 28 Lipschitz mappings, 87 lower bounds, 3, 50 lower limits, 53 lower sums, 91, 92

mappings, 60 metric spaces, 1 metrics, 1 modulus, 34 monotone sequences, 36 monotonically increasing functions, 74

one-sided limits, 73 open balls, 4 open coverings, 9 open intervals, 5 open sets, 4

partitions of intervals, 91 common refinements, 95 refinements, 93

path connected sets, 76 perfect sets, 24 pointwise convergence, 67 power series, 46 product rule, 81

Q, 3, 23 quotient rule, 81

R, 1 R^{n}, 2 radius of convergence, 47 ratio test, 56 real part, 34 rearrangements, 57 relatively open sets, 13 Riemann integrability, 96 Riemann integrals, 93, 96 Riemann sums, 91 Riemann–Stieltjes integrability, 96 Riemann–Stieltjes integrals, 94, 96 Riemann–Stieltjes sums, 92 root test, 55

separable metric spaces, 25

separated sets, 31

sequential compactness, 40 standard metric on C, 35 subsequences, 39 supremum, 2, 50 supremum metric, 70

totally bounded sets, 21 triangle inequality, 1

uncountable sets, 24 uniform continuity, 65 uniform convergence, 67 unit step function, 75 upper bounds, 2, 50 upper limits, 53 upper sums, 91, 92

Weierstrass’ criterion, 68 Weierstrass’ theorem, 79

Z, 22 Z_{+}, 9