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REAL VARIABLES with BASIC METRIC SPACE TOPOLOGY Robert B. Ash Department of Mathematics University of Illinois at Urbana-Champaign © Copyright 2007 by Robert B. Ash. Paper or electronic copies for noncommercial use may be made freely without explicit permission of the author. All other rights are reserved, 1.1 Basic Terminology 1.2 Finite and Infinite Sets; Countably Infinite and Uncountably Infinite Sets 1.3 Distance and Convergence 1.4 Minicourse in Basic Logic 1.5 Limit Points and Closure Review Problems for Chapter 1 2.1 Unions and Intersections of Open and Closed Sets 25 2.2 Compactness 28 Vi CONTENTS 2.3 Some Applications of Compactness 33 2.4 Least Upper Bounds and Completeness Review Problems for Chapter 2 42 3.1 Generalization of the Limit Concept 3.2 Some Properties of Upper and Lower Limits 3.3 Convergence of Power Series Review Problems for Chapter 3 4.1 Continuity: Ideas, Basic Terminology, Properties 4.2 Continuity and Compactness 4.3 Types of Discontinuities 4.4 The Cantor Set Review Problems for Chapter 4 DIFFER 5.1 The Derivative and Its Basic Properties 5.2 Additional Properties of the Derivative; Some Applications of the Mean Value Theorem R eview Problems for Chapter 5 6.1 Definition of the Integral 6.2 Properties of the Integral 6.3 Functions of Bounded Variation 6.4 Some Useful Integration Theorems Review Problems for Chapter 6 CONTENTS vil 7.1 Pointwise and Uniform Convergence 7.2 Uniform Convergence and Limit Operations 7.3 The Weierstrass M-test and Applications 7.4 Equicontinuity and the Arzela—Ascoli Th 7.5 The Weierstrass Approximation Theorem Review Problems for Chapter 7 8.1 The Extension Problem 8.2 Baire Category Theorem 8.3 Connectedness 8.4 Semicontinuous Functions Review Problems for Chapter 8 9.1 Some Compactness Results < eplacing Cantor’s Nested Set P roperty The veal Numbers R evisited This is a text for a first course in real variables. The subject matter is fundamental for more advanced mathematical work, specifically in the areas of complex variables, measure theory, differential equations, functional analysis, and probability. In addition, many students of engineering, physics, and economics find that they need to know real analysis in order to cope with the professional literature in their fields. Standard mathematical writing, with its emphasis on formalism and abstraction, tends to create barriers to learning and focus on minor technical details at the expense of intuition. On the other hand, a certain amount of abstraction is unavoidable if one is to give a sound and coherent presentation. This book attempts to strike a balance that will reach the widest audience possible without sacrificing precision. (My original training was in electrical engineering, and I later became a mathematician. I seem to be able to reach students whose prior exposure to abstract reasoning is limited.) The most important skill a student must develop in order to learn any area of mathematics is the ability to think intuitively, and a text should encourage this process—not hinder it. I find it useful to consider con- crete examples that have all the features of the general case under consideration, to draw diagrams whenever appropriate, and to give geometric or physical interpretation of results. I rely especially on one ix X PREFACE of the most useful of all learning devices: the inclusion of detailed solutions to exercises. Solutions to problems are commonplace in ele- mentary texts but quite rare (although equally valuable) at the upper division undergraduate and graduate level. This feature makes the book suitable for independent study, and further widens the audience. I have normally covered the first seven chapters in a one-semester course. The subject matter includes metric spaces, Euclidean spaces and their basic topological properties, sequences and series of real numbers, continuous functions, differentiation, Riemann-Stieltjes integration, and uniform convergence and applications. Chapters 8 and 9 contain additional topological results, and various sections might be assigned in connection with special student projects or serve as a nice introduction to a full course in general topology. The subject matter covered in basic courses in real variables is standard, almost canonical, but there is still room for innovation in the presentation. I have taken Cantor’s Nested Set Property, rather than Cauchy completeness or least upper bound completeness, as the basic topological axiom for the real numbers. This allows a significant simplification of the proofs of the basic topological properties of R’. 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. Th e 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, 5.237999 - +. = 5.2380000.... 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. 2 INTRODUCTION Ch | B, denoted by A U B, is the set of points belonging to either A or | B (or both: from now on, the word “or” al- ways has the so-called inclusive connotation “or both” unless other- wise specified). T7 h e intersection of two sets A and B , denoted by A N B, is the set of points belonging to both A and B. The complement of a set A, denoted by A‘, is the set of points not belonging to A. (Generally, we will be working in a fixed space 0 (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 2, and thus A® is the set of points of 0 that do not belong to A). Unions, intersections, and complements may be represented by pic- tures 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 A;, A2,..., denoted by Aj UA2U... or by U, Ai, is the set of points belonging to at least one of the Aj; the intersection of Aj, A2,..., denoted by Aj M A2N... or by (); Aj, is the set of points belonging to all the A;. The unionn of sets A;,...,A, is often written as L_, A; , and the union of an infinite sequence A,, A2,.. is denoted by (J%°, Ai, 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 setsA , B, C, (1) AN(BUC)=(ANB)U(ANC) (the word “distributive” is used because in this formula, at least, intersection behaves like ordinary multiplication and union like addition). AUB . ANB Figure 1.1.1 Union, Intersection, and Complement Q INTRODUCTION 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”.) Ifx € AN(BUC), thenx € A andx € BUC,sothatx € Bor xEC. Case 1. x € Aandx € B ; thenx € AM B;hencex € (AN B)U(ANC). Case 2, x €Aandx € C; thenx € ANC; hencex € (AN 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 C 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 C E but D # E, we say that D is a proper subset of E', sometimes written D C 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 NB) U (ANC). Th enx € AN Borx € ANC; thus, we kn x €A,andalsox € Borx € C, as desired. Figure 1.1.2 Distributive AN(BU C)=(AN B) U (ANC) Law