Real Variables with Basic Metric Space Topology

Real Variables with Basic Metric Space Topology

Contents

1.Introduction

1.1BASIC TERMINOLOGY

1.1.1Definitions and Comments

1.2FINITE AND INFINITE SETS; COUNTABLY INFINITE AND UNCOUNTABLY INFINITE SETS

1.3DISTANCE AND CONVERGENCE

1.3.1Definitions and Comments

1.4MINICOURSE IN BASIC LOGIC

1.4.1Truth Tables

1.4.2Types of Proof

1.4.3Quantifiers

1.4.4Mathematical Induction

1.4.5Negations

1.5LIMIT POINTS AND CLOSURE

2.Some Basic Topological Properties of Rp

2.1UNIONS AND INTERSECTIONS OF OPEN AND CLOSED SETS

2.2COMPACTNESS

2.2.1Definition

2.2.2Nested Set Property

2.2.3Definition

2.2.5 Heine-Borel Theorem

2.3SOME APPLICATIONS OF COMPACTNESS

2.3.2Bolzano–Weierstrass Theorem

2.4LEAST UPPER BOUNDS AND COMPLETENESS

2.4.1Definitions

2.4.4Definitions and Comments

3.Upper and Lower Limits of Sequences of Real Numbers

3.1GENERALIZATION OF THE LIMIT CONCEPT

3.1.1Definitions and Comments

3.2SOME PROPERTIES OF UPPER AND LOWER LIMITS

3.2.1Definitions and Comments

3.2.4Remark

3.3CONVERGENCE OF POWER SERIES

4.Continuous Functions

4.1CONTINUITY: IDEAS, BASIC TERMINOLOGY, PROPERTIES

4.1.1Definition

4.1.4Definition

4.1.7Definition

4.2CONTINUITY AND COMPACTNESS

4.2.3Definition

4.3TYPES OF DISCONTINUITIES

4.3.2Intermediate Value Theorem

4.4THE CANTOR SET

4.4.2Remarks

5.Differentiation

5.1THE DERIVATIVE AND ITS BASIC PROPERTIES

5.1.1Definition and Comments

5.1.4Mean Value Theorem

5.1.6Generalized Mean Value Theorem

5.2ADDITIONAL PROPERTIES OF THE DERIVATIVE; SOME APPLICATIONS OF THE MEAN VALUE THEOREM

5.2.1Intermediate Value Theorem for Derivatives

5.2.3L’Hospital’s Rule

5.2.4Taylor’s Formula with Remainder

6.Riemann-Stieltjes Integration

6.1DEFINITION OF THE INTEGRAL

6.1.1Definitions and Comments

6.1.3Definition

6.2PROPERTIES OF THE INTEGRAL

6.2.3Evaluation Formula

6.2.4Fundamental Theorem of Calculus

6.3FUNCTIONS OF BOUNDED VARIATION

6.3.1Definitions and Comments

6.4SOME USEFUL INTEGRATION THEOREMS

6.4.1Integration by Parts

6.4.2Change of Variable Formula

6.4.3Mean Value Theorem for Integrals

6.4.4Upper Bounds on Integrals

6.4.5Improper Integrals

7.Uniform Convergence and Applications

7.1POINTWISE AND UNIFORM CONVERGENCE

7.1.1Examples of Invalid Interchange of Operations

7.1.2Definitions

7.1.3Example

7.2UNIFORM CONVERGENCE AND LIMIT OPERATIONS

7.2.5Dini's Theorem

7.3THE WEIERSTRASS M-TEST AND APPLICATIONS

7.3.1Weierstrass M-Test

7.3.2Example

7.3.3An Everywhere Continuous, Nowhere Differentiate Function

7.4 EQUICONTINU1TY AND THE ARZELA-ASCOU THEOREM

7.4.3Definition

7.4.4Arzela-Ascoli Theorem

7.5THE WEIERSTRASS APPROXIMATION THEOREM

7.5.3Weierstrass Approximation Theorem

8.Further Topological Results

8.1THE EXTENSION PROBLEM

8.1.3Tietze Extension Theorem

8.2BAIRE CATEGORY THEOREM

8.2.1Definitions and Comments

8.2.2Baire Category Theorem

8.3CONNECTEDNESS

8.3.1Definitions

8.4SEMICONTINUOUS FUNCTIONS

8.4.1Definitions and Comments

9.Epilogue

9.1SOME COMPACTNESS RESULTS

9.1.1Definitions and Comments

9.2REPLACING CANTORS NESTED SET PROPERTY

9.3THE REAL NUMBERS REVISITED

Solutions to Problems

1

INTRODUCTION

1.1BASIC TERMINOLOGY

In a course in real analysis, the normal procedure is to begin with a definition of the real numbers, either by means of a set of axioms or by a constructive procedure which starts with the God–given set of positive integers. The set of all integers is constructed, and from this the rational numbers are obtained, and finally the reals. A discussion of this type is part of the area of logic and foundations rather than real analysis, and we will postpone it until much later. For now we’ll take the point of view that we know what the real numbers are: A real number is an integer plus an infinite decimal, for example, 65.7204…

If the decimal ends in all nines, we have two representations of the same real number, for example,

We will often talk about sets of real numbers, and therefore a modest amount of set–theoretic terminology is necessary before we can get anywhere. You have probably seen most of this in another course, so we will proceed rather quickly.

1.1.1Definitions and Comments

The union of two sets A and B, denoted by A ∪ B, is the set of points belonging to either A or B (or both; from now on, the word or always has the so–called inclusive connotation or both unless otherwise specified).

The intersection of two sets A and B, denoted by A ∩ B, is the set of points belonging to both A and B.

The complement of a set A, denoted by Ac, is the set of points not belonging to A. (Generally, we will be working in a fixed space Ω (sometimes called the universe), for example, the set of real numbers or perhaps the set of pairs of real numbers, that is, the Euclidean plane. All sets under discussion will consist of various points of Ω, and thus Ac is the set of points of Ω that do not belong to A).

Unions, intersections, and complements may be represented by pictures called Venn diagrams that are probably familiar to many readers; see Fig. 1.1.1.

Unions and intersections may be defined for more than two sets, in fact for an arbitrary collection of sets.

The union of sets A1, A2,…, denoted by A1 ∪ A2 ∪ … or by , is the set of points belonging to at least one of the A1; the intersection of A1, A2,…, denoted by A1 ∩ Α2 ∩ … or by , is the set of points belonging to all the Ai. The union of sets A1, …, An is often written as , and the union of an infinite sequence A1, A2, … is denoted by , with similar notation for intersection.

There are a few identities involving unions, intersections, and complements that come up occasionally. For example, the distributive law holds: for arbitrary sets A, B, C,

(the word distributive is used because in this formula, at least, intersection behaves like ordinary multiplication and union like addition).

Figure 1.1.1 Union, Intersection, and Complement

The formula may be verified by drawing a Venn diagram (Fig. 1.1.2) or by showing that the sets on the left and right sides of the equality have the same members, as follows. (The symbol ∈ means belongs to.) If x ∈ A ∩ (B ∪ C), then x ∈ A and x ∈ B ∪ C, so that x ∈ B or x ∈ C.

Case 1. x ∈ A and x ∈ B; then x ∈ A ∩ B; hence x ∈ (A ∩ B) ∪ (A ∩ C).

Case 2. x ∈ A and x ∈ C ; then x ∈ A ∩ C; hence x ∈ (A ∩ B) ∪ (A ∩ C).

Caution. So far we have shown only that the set on the left is a subset of the set on the right. In general, the set D is said to be a subset of the set E if every point of D also belongs to E (see Fig. 1.1.3). We use the notation D ⊆ E ; sometimes we say that E contains D or the D is contained in E. In order to show that D = E we must also show that every point of E belongs to D. If D ⊆ E but D ≠ E, we say that D is a proper subset of E, sometimes written D ⊂ E. Note that according to the definition, a set is a subset of itself.

To return to the proof of the distributive law, suppose x ∈ (A ∩ B) ∪ (A ∩ C). Then x ∈ A ∩ B or x ∈ A ∩ C; thus, we know definitely that x ∈ A, and also x ∈ B or x ∈ C, as desired.

Figure 1.1.2 Distributive Law

Figure 1.1.3 Subset Relation

The following identities, called the De Morgan laws, are often useful:

As above, the identities can be verified by a Venn diagram or a formal argument using the definitions of union, intersection, and complement. In fact the De Morgan laws can be extended to an arbitrary collection of sets, as follows:

The Venn diagram approach is not useful with such a large collection of sets, and the formal method must be used (Problem 1).

To complete the necessary set-theoretic terminology, the empty set (set with no members) will be denoted by . If A is any set, then . (If you produce an element , I will be delighted to show that x belongs to A.) We will look at this idea in more detail in Section 1.4. The sets Ai are said to be disjoint or mutually exclusive if there is no overlap between any pair of sets; that is,

As an example, let A = {1, 2, 3}, B = {1, 2}, C = {3, 5} (the notation is standard: a set is described by listing its members, so that, for example, A is the set consisting of the numbers 1, 2, and 3). In this case, A, B, C are not disjoint, although . For disjointness we must have

, and here we have

. Since , we may say that B and C are disjoint.

The set–theoretic difference between A and B is defined by

Problems for Section 1.1

1.Prove the De Morgan laws for an arbitrary collection of sets (Eq. (3)).

2.Prove that union distributes over intersection; i.e.,

3.Show that the union of three arbitrary sets can be written as a disjoint union in the following way:

4.Continuing Problem 3, if A1, A2, … are arbitrary sets, show that

5.If A, B, C, D are arbitrary sets, express the set of elements belonging to at least two of the sets A, B, C, D, using unions and intersections of A, B, C, D.

6.Repeat Problem 5 with at least two replaced by exactly two and use complements as well as unions and intersections.

1.2FINITE AND INFINITE SETS; COUNTABLY INFINITE AND UNCOUNTABLY INFINITE SETS

Sometimes we need to know something about the size of a set, in particular, whether it is finite. If a set is infinite, it may be useful to know if it can somehow be counted or if it is uncountable. The definitions are as follows.

A finite set is one that can be put in one–to–one correspondence with {1, 2,…, n} for some positive integer n (by convention, the empty set is regarded as finite). An infinite set is a set that is not finite. A countably infinite set is one that can be put in one–to–one correspondence with the entire set of positive integers. This means simply that the points can be labeled 1, 2, 3,…. A set is uncountably infinite if it is infinite but not countably infinite. It is convenient to call a set countable if it is either finite or countably infinite; thus, uncountable is synonymous with uncountably infinite.

Possibly you have seen the classic arguments that the set of rational numbers is countably infinite, but the set of all real numbers is uncountably infinite. To count the positive rationals, we devise an explicit scheme (see Fig. 1.2.1). The procedure amounts to counting the points of an infinite rectangular array, and is slightly inefficient because each rational number appears infinitely often: for example, 1/2 = 2/4 = 3/6, and so on. After the first appearance of a number (say r2 = 1/2 in Fig. 1.2.1), all other appearances are skipped in making the count. To show that the reals are uncountable, we use the Cantor diagonal process. (This idea will occur several times later on.) Suppose we were able to count the real numbers between 0 and 1; list the numbers, in decimal form, as follows:

Figure 1.2.1 Counting the Rational Numbers

Then form the real number r = .b1b2b3 …, where b1 ≠ a11, b2 ≠ a22, b3 ≠ a33, … To avoid the ambiguity caused by expansions ending in all nines or all zeros, we can, if we like, take 1 ≤ bn ≤ 8 for all n. Then r is a real number between 0 and 1, but cannot appear on the list.

There is an extensive theory of infinite sets, but we will be content with a few basic results.

1.2.1THEOREM. There are 2n subsets of {1, 2,…, n}.

Proof. If S ⊆ {1, 2, …, n}, then 1 ∈ S or 1 ∉ S (two choices),…, n ∈ S or n ∉ S (two choices). The total number of subsets is the same as the total number of choices, namely, 2 × 2 × ··· × 2 = 2n. ■

1.2.2THEOREM. There are uncountably many subsets of the positive integers.

Proof. Make a correspondence between subsets of the positive integers and binary representations of real numbers between 0 and 1, as follows:

Since there are uncountably many reals between 0 and 1, there are uncountably many subsets of {1, 2,…}.¹ ■

1.2.3THEOREM. A countable union of countable sets is countable. In other words, if for each n = 1, 2, …, An is countable, then is countable.

Proof. List the members of the Ai as follows:

Then count by the same procedure we used to count the rationals. ■

Problems for Section 1.2

1.Give an alternative proof of Theorem 1.2.2, as follows. If S1, S2, … is a list of subsets of {1, 2,…}, construct a subset S that cannot possibly be on the list.

2.Show that there are only countably many finite subsets of the positive integers.

3.Verify informally that the mapping

is a one-to-one correspondence between ordered pairs (x, y) of nonnegative integers and nonnegative integers (see diagram).

4.The method of Fig. 1.2.1 shows that the positive rationals are countably infinite. How would you modify the procedure so as to count all the rationals?

5.Suppose that the rational numbers between 0 and 1 are listed as in (4). We then pick a rational r = .r1r2r3 … with rn ≠ ann, n = 1, 2,…. Why doesn’t this show that the rationals are uncountable?

6.Show that it is impossible to list the rational numbers in increasing order.

7.Show that for any positive integer n the set of all (x1,…, xn), where the xi are rational, is countable.

1.3DISTANCE AND CONVERGENCE

One of the basic ideas of analysis is that of convergence; a sequence of numbers xn converges to a number x if, as n gets very large, xn gets very close to x ; in other words, the distance between xn and x gets very small. The key concept is that of distance; as long as we have a distance function, we can talk about convergence. You are familiar with several distance functions. If x and y are points on the real line, the distance between them is |x – y|; if x = (x1, x2) and y = (y1, y2) are points in the plane, the Euclidean distance between them is [(x1 – y1)² + (x2 – y2)²]¹/². What properties must a distance function satisfy in order that we can talk about convergence sensibly? It turns out that only a few are needed, as follows.

1.3.1Definitions and Comments

A metric or distance function on a set Ω is an assignment, to each pair of points (x, y), x, y ∈ Ω, of a nonnegative real number d (x, y), such that for all x, y, z ∈ Ω we have

Statement (c) is called the triangle inequality; if x, y, and z are vertices of a triangle in the plane, (c) says that the length of one side of the triangle cannot exceed the sum of the lengths of the other two sides.

A set Ω on which a distance function is defined is called a metric space. Our basic metric spaces will be the set of real numbers, to be denoted from now on by R, and Euclidean p–space Rp, the set of all p–tuples (x1,…, xp) of real numbers. The metric on Rp is given by

When p = 1, we have Rp = R, d(x, y) = |x – y|. We know because of our familiarity with elementary geometry that d is a metric when p ≤ 3, but this must be proved when p > 3. However, it’s probably best to wait until we