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Introduction to Real Analysis

INTRODUCTION TO REAL ANALYSISThird EditionRobert G. BartleDonald R. SherbertEastern Michigan University, YpsilantiUniversity of Illinois, Urbana-ChampaignJohn Wiley & Sons, Inc.

ACQUISITION EDITORBarbara HollandASSOCIATE EDITORSharon PrendergastPRODUCTION EDITORKen SantorPHOTO EDITORNicole HorlacherILLUS TRATION COORDINATORGene AielloThis book was set in Times Roman by Eigentype Compositors, and printed and bound by HamiltonPrinting Company. The cover was printed by Phoenix Color Corporation.This book is printed on acid-free paper. §The paper in this book was manufactured by a mill whose forest management programs include sustained yieldharvesting of its timberlands. Sustained yield harvesting principles ensure that the numbers of trees CUI each yeardoes not exceed the amount of new growth.Copyright 2000 John Wiley & Sons, Inc. All rights reserved.No part of this publication may be reproduced, stored in a retrieval system or transmittedin any form or by any means, electronic, mechanical, photocopying, recording, scanningor otherwise, except as permitted under Section 107 or 108 of the 1976 United StatesCopyright Act, without either the prior written permission of the Publisher, or authorizationthrough payment of the appropriate per-copy fee to the Copyright Clearance Center,222 Rosewood Drive, Danvers, MA 01923, (508) 750-8400, fax (508) 750-4470.Requests to the Publisher for permission should be addressed to the Permissions Department,John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012,(212) 850-601 1, fax (212) 850-6008, E-Mail: PERMRE@W1LEY.COM.Library of Congress Cataloging in Publication Data:Bartle, Robert Gardner, 1927Introduction to real analysis / Robert G . Bartle, Donald R., Sherbert. - 3rd bibliographical references and index.ISBN 0-47 1-32148-6 (a1k. paper)1. Mathematical analysis. 2. Functions of real variables.1. Sherbert, Donald R., 1935- . II. Title.QA300.B294515-dc21200099-13829CIPA. M. S. Classification 26-01Printed in the United States of America201918171615141312II

PREFACEThe study of real analysis is indispensible for a prospective graduate student of pure orapplied mathematics. It also has great value for any undergraduate student who wishesto go beyond the routine manipulations of formulas to solve standard problems, becauseit develops the ability to think deductively, analyze mathematical situations, and extendideas to a new context. In recent years, mathematics has become valuable in many areas,including economics and management science as well as the physical sciences, engineering,and computer science. Our goal is to provide an accessible, reasonably paced textbook inthe fundamental concepts and techniques of real analysis for students in these areas. Thisbook is designed for students who have studied calculus as it is traditionally presented inthe United States. While students find this book challenging, our experience is that seriousstudents at this level are fully capable of mastering the material presented here.The first two editions of this book were very well received, and we have taken painsto maintain the same spirit and user-friendly approach. In preparing this edition, we haveexamined every section and set of exercises, streamlined some arguments, provided a fewnew examples, moved certain topics to new locations, and made revisions. Except for thenew Chapter 10, which deals with the generalized Riemann integral, we have not addedmuch new material. While there is more material than can be covered in one semester,instructors may wish to use certain topics as honors projects or extra credit assignments.It is desirable that the student have had some exposure to proofs, but we do not assumethat to be the case. To provide some help for students in analyzing proofs of theorems,we include an appendix on "Logic and Proofs" that discusses topics such as implications,quantifiers, negations, contrapositives, and different types of proofs. We have kept thediscussion informal to avoid becoming mired in the technical details of formal logic. Wefeel that it is a more useful experience to learn how to construct proofs by first watchingand then doing than by reading about techniques of proof.We have adopted a medium level of generality consistently throughout the book: wepresent results that are general enough to cover cases that actually arise, but we do not strivefor maximum generality. In the main, we proceed from the particular to the general. Thuswe consider continuous functions on open and closed intervals in detail, but we are carefulto present proofs that can readily be adapted to a more general situation. (In Chapter 1 1we take particular advantage of the approach.) We believe that it is important to providestudents with many examples to aid them in their understanding, and we have compiledrather extensive lists of exercises to challenge them. While we do leave routine proofs asexercises, we do not try to attain brevity by relegating difficult proofs to the exercises.However, in some of the later sections, we do break down a moderately difficult exerciseinto a sequence of steps.In Chapter 1 we present a brief summary of the notions and notations for sets andfunctions that we use. A discussion of Mathematical Induction is also given, since inductiveproofs arise frequently. We also include a short section on finite, countable and infinite sets.We recommend that this chapter be covered quickly, or used as background material,returning later as necessary.v

viPREFACEChapter 2 presents the properties of the real number system lR. The first two sectionsdeal with the Algebraic and Order Properties and provide some practice in writing proofsof elementary results. The crucial Completeness Property is given in Section 2.3 as theSupremum Property, and its ramifications are discussed throughout the remainder of thischapter.In Chapter 3 we give a thorough treatment of sequences in IR and the associated limitconcepts. The material is of the greatest importance; fortunately, students find it rathernatural although it takes some time for them to become fully accustomed to the use of .In the new Section 3.7, we give a brief introduction to infinite series, so that this importanttopic will not be omitted due to a shortage of time.Chapter 4 on limits of functions and Chapter 5 on continuous functions constitutethe heart of the book. Our discussion of limits and continuity relies heavily on the use ofsequences, and the closely parallel approach of these chapters reinforces the understandingof these essential topics. The fundamental properties of continuous functions (on intervals)are discussed in Section 5.3 and 5.4. The notion of a "gauge" is introduced in Section 5.5and used to give alternative proofs of these properties. Monotone functions are discussedin Section 5.6.The basic theory of the derivative is given in the first part of Chapter 6. This importantmaterial is standard, except that we have used a result of Caratheodory to give simplerproofs of the Chain Rule and the Inversion Theorem. The remainder of this chapter consistsof applications of the Mean Value Theorem and may be explored as time permits.Chapter 7, dealing with the Riemann integral, has been completely revised in thisedition. Rather than introducing upper and lower integrals (as we did in the previouseditions), we here define the integral as a limit of Riemann sums. This has the advantage thatit is consistent with the students ' first exposure to the integral in calculus and in applications;since it is not dependent on order properties, it permits immediate generalization to complex and vector-valued functions that students may encounter in later courses. Contrary topopular opinion, this limit approach is no more difficult than the order approach. It also isconsistent with the generalized Riemann integral that is discussed in detail in Chapter 10.Section 7.4 gives a brief discussion of the familiar numerical methods of calculating theintegral of continuous functions.Sequences of functions and uniform convergence are discussed in the first two sec tions of Chapter 8, and the basic transcendental functions are put on a firm foundation inSection 8.3 and 8.4 by using uniform convergence. Chapter 9 completes our discussion ofinfinite series. Chapters 8 and 9 are intrinsically important, and they also show how thematerial in the earlier chapters can be applied.Chapter 10 is completely new; it is a presentation of the generalized Riemann integral(sometimes called the "Henstock-Kurzweil" or the "gauge" integral). It will be new to manyreaders, and we think they will be amazed that such an apparently minor modification ofthe definition of the Riemann integral can lead to an integral that is more general than theLebesgue integral. We believe that this relatively new approach to integration theory is bothaccessible and exciting to anyone who has studied the basic Riemann integral.The final Chapter 1 1 deals with topological concepts. Earlier proofs given for intervalsare extended to a more abstract setting. For example, the concept of compactness is givenproper emphasis and metric spaces are introduced. This chapter will be very useful forstudents continuing to graduate courses in mathematics.Throughout the book we have paid more attention to topics from numerical analysisand approximation theory than is usual. We have done so because of the importance ofthese areas, and to show that real analysis is not merely an exercise in abstract thought.

PREFACEviiWe have provided rather lengthy lists of exercises, some easy and some challenging.We have provided "hints" for many of these exercises, to help students get started toward asolution or to check their "answer". More complete solutions of almost every exercise aregiven in a separate Instructor' s Manual, which is available to teachers upon request to thepublisher.It is a satisfying experience to see how the mathematical maturity of the studentsincreases and how the students gradually learn to work comfortably with concepts thatinitially seemed so mysterious. But there is no doubt that a lot of hard work is required onthe part of both the students and the teachers.In order to enrich the historical perspective of the book, we include brief biographicalsketches of some famous mathematicians who contributed to this area. We are particularlyindebted to Dr. Patrick Muldowney for providing us with his photograph of ProfessorsHenstock and Kurzweil. We also thank John Wiley & Sons for obtaining photographs ofthe other mathematicians.We have received many helpful comments from colleagues at a wide variety of in stitutions who have taught from earlier editions and liked the book enough to expresstheir opinions about how to improve it. We appreciate their remarks and suggestions, eventhough we did not always follow their advice. We thank them for communicating with usand wish them well in their endeavors to impart the challenge and excitement of learningreal analysis and "real" mathematics. It is our hope that they will find this new edition evenmore helpful than the earlier ones.February 24, 1999Ypsilanti and UrbanaRobert G. BartleDonald R. SherbertTHE GREEK ALPHABETABr EZHeIKAMexfJy/)e IotaKappaLambdaMuN 0IlPI;T1 I X\IIQv hiPsiOmega

To our wives, Carolyn and Janice,with our appreciation for theirpatience, support, and love.

CONTENTSCHAPTER 1PRELIMINARIES 1CHAPTER 2THE REAL NUMBERS 22CHAPTER 3CHAPTER 4C.HAPTER 51 . 1 Sets and Functions· 11.2 Mathematical Induction 121 .3 Finite and Infinite Sets Algebraic and Order Properties of IRAbsolute Value and Real Line 31The Completeness Property of IR 34Applications of the Supremum PropertyIntervals 442238SEQUENCES AND SERIES 523. and Their Limits 53Limit Theorems 60Monotone Sequences 68Subsequences and the Bolzano-Weierstrass TheoremThe Cauchy Criterion 80Properly Divergent Sequences 86Introduction to Infinite Series 8975LIMITS 964. 1 Limits of Functions 974.2 Limit Theorems 1054.3 Some Extensions of the Limit Concept111CONTINUOUS FUNCTIONS 1195. Functions 120Combinations of Continuous FunctionsContinuous Functions on Intervals 129Uniform Continuity 136Continuity and Gauges 145Monotone and Inverse Functions 149125ix

xCONTENTSCHAPTER 6CHAPTER 7CHAPTER 8CHAPTER 9DIFFERENTIATION 1576. Derivative 158The Mean Value TheoremL'Hospital's Rules 176Taylor' s Theorem 183168THE RIEMANN INTEGRAL 1937. Riemann Integral 194Riemann Integrable Functions 202The Fundamental Theorem 210Approximate Integration 219SEQUENCES OF FUNCTIONS 2278. and Uniform Convergence 227Interchange of Limits 233The Exponential and Logarithmic FunctionsThe Trigonometric Functions 246239INFINITE SERIES 2539. Convergence 253Tests for Absolute Convergence 257Tests for Nonabsolute Convergence 263Series of Functions 266CHAPTER 10THE GENERALIZED RIEMANN INTEGRAL 274CHAPTER 11A GLIMPSE INTO TOPOLOGY 31210.110.210.310.41 1. 11 1.21 1.31 1.4Definition and Main Properties 275Improper and Lebesgue Integrals 287Infinite Intervals 294Convergence Theorems 301Open and Closed Sets in lRCompact Sets 319Continuous Functions 323Metric Spaces 327312APPENDIX ALOGIC AND PROOFS 334APPENDIX BFINITE AND COUNTABLE SETS 343APPENDIX CTHE RIEMANN AND LEBESGUE CRITERIA 347


CHAPTER 1PRELIMINARIESIn this initial chapter we will present the background needed for the study of real analysis.Section 1.1 consists of a brief survey of set operations and functions, two vital tools for allof mathematics. In it we establish the notation and state the basic definitions and propertiesthat will be used throughout the book. We will regard the word "set" as synonymous withthe words "class", "collection", and "family", and we will not define these terms or give alist of axioms for set theory. This approach, often referred to as "naive" set theory, is quiteadequate for working with sets in the context of real analysis.Section 1 .2 is concerned with a special method of proof called Mathematical Induction.It is related to the fundamental properties of the natural number system and, though it isrestricted to proving particular types of statements, it is important and used frequently. Aninformal discussion of the different types of proofs that are used in mathematics, such ascontrapositives and proofs by contradiction, can be found in Appendix A.In Section 1 .3 we apply some of the tools presented in the first two sections of thischapter to a discussion of what it means for a set to be finite or infinite. Careful definitionsare given and some basic consequences of these definitions are derived. The importantresult that the set of rational numbers is countably infinite is established.In addition to introducing basic concepts and establishing terminology and notation,this chapter also provides the reader with some initial experience in working with precisedefinitions and writing proofs. The careful study of real analysis unavoidably entails thereading and writing of proofs, and like any skill, it is necessary to practice. This chapter isa starting point.Section 1.1Sets and FunctionsIn this section we give a brief review of the terminology and notation thatwill be used in this text. We suggest that you look through quickly and come back laterwhen you need to recall the meaning of a term or a symbol.If an element is in a set A, we writeTo the reader:xand say that is a member of A, or thatxX EAxxbelongs to A . If x is A.not in A, we writeIf very element of a set A also belongs to a set B , we say that A is a subset of B and writeorWe say that a set A is a proper subset of a set B if A B, but there is at least one elementof B that is not in A. In this case we sometimes writeA C B.1

2CHAPTER 1PRELIMINARlES1.1.1 Definition Two sets A and B are said to be equal. and we write A B. if theycontain the same elements.Thus. to prove that the sets A and B are equal. we must show thatA B and B A.A set is normally defined by either listing its elements explicitly. or by specifying aproperty that determines the elements of the set. If P denotes a property that is meaningfuland unambiguous for elements of a set S. then we write{x E S P(x)}:for the set of all elements x in S for which the property P is true. If the set S is understoodfrom the context. then it is often omitted in this notation.Several special sets are used throughout this book. and they are denoted by standardsymbols. (ytle will use the symbol : to mean that the symbol on the left is being definedby the symbol on the right.)The set of natural numbers N : {I. 2. 3 }.The set of integers Z : to. 1. -1.2, -2, · · . },The set ofrational numbers Q : {min : m, n E Z and n I- OJ.The set of real numbers RThe set lR of real numbers is of fundamental importance for us and will be discussedat length in Chapter 2. . . . 1.1.2 Examples (a)The set{x E N : x2 - 3x 2 O}consists of those natural numbers satisfying the stated equation. Since the only solutions ofthis quadratic equation are x 1 and x 2, we can denote this set more simply by {I, 2}.(b) A natural number n is even if it has the form n 2k for some k E N. The set of evennatural numbers can be written{2k : k E N},which is less cumbersome than {n E N : n 2k, k E N}. Similarly, the set of odd naturalnumbers can be written{2k - 1 k E N}.:oSet OperationsWe now define the methods of obtaining new sets from given ones. Note that these setoperations are based on the meaning of the words "or", "and", and "not". For the union,it is important to be aware of the fact that the word "or" is used in the inclusive sense,allowing the possibility that x may belong to both sets. In legal terminology, this inclusivesense is sometimes indicated by "andlor".1.1.3 Definition(a)The union of sets A and B is the setAU B : {x : x E A or x E B} .

1.1 SETS AND FUNCTIONS3(b) The intersection of the sets A and B is the setAnB: {x : x E A and x E B} .(c) The complement of B relative to A is the setA\B: {x : x E A and xA U B !IIID(a) A U BFigure 1.1.1rt(b) A n BB} .(c) A\BA\B The set that has no elements is called the empty set and is denoted by the symbol 0.Two sets A and B are said to be disjoint if they have no elements in common; this can beexpressed by writing A n B 0.To illustrate the method of proving set equalities, we will next establish one of thefor three sets. The proof of the other one is left as an exercise.DeMorgan laws1.1.4 TheoremIf A, B, Care sets, then(a) A\(B U C) (A\B) n (A\C),(b) A\(B n C) (A\B) U (A\C) .To prove (a), we will show that every element in A\ (B U C) is contained in both(A\B) and (A \C), and conversely.If x is in A\(B U C), then x is in A, but x is not in B U C . Hence x is in A, but xis neither in B nor in C . Therefore, x is in A but not B, and x is in A but not C . Thus,x E A\B and x E A\C, which shows that x E (A \B) n (A\C).Conversely, if x E (A\B) n (A\C), then x E (A\B) and x E (A\C). Hence x E Aand both x rt B and x rt C . Therefore, x E A and x rt (B U C), so that x E A\ (B U C) .Since the sets (A\B) n (A\ C) and A\ (B U C) contain the same elements, they areProof.equal by Definition1.1.1.Q.E.D.There are times when it is desirable to form unions and intersections of more than twosets. For a finite collection of sets {A I ' A 2 , . . , An }, their union is the set A consisting ofall elements that belong toof the sets Ak , and their intersection consists of allel ments that belong to of the sets Ak This is extended to an infinite collection of sets {A I' A2 , , An ' . . .} as follows. Theirunion is the set of

Introduction to real analysis / Robert G. Bartle, Donald R., Sherbert. -3rd ed. p. cm. Includes bibliographical references and index. ISBN 0-471-32148-6 (a1k. paper) 1. Mathematical analysis. 2. Functions of real variables. 1. Sherbert, Donald R., 1935- . II. Title. QA300.B294 2000 515-dc21 A. M. S. Classification 26-01 Printed in the United States of America 20 19 18 17 16 15 14 13 12 II 99 .

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