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FFIRS12/15/201010:13:22Page 1Introduction to Real Analysis

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FFIRS12/15/201010:13:22Page 3INTRODUCTION TO REAL ANALYSISFourth EditionRobert G. BartleDonald R. SherbertUniversity of Illinois, Urbana-ChampaignJohn Wiley & Sons, Inc.

FFIRS12/15/201010:13:22Page 4VP & PUBLISHERPROJECT EDITORMARKETING MANAGERMEDIA EDITORPHOTO RESEARCHERPRODUCTION MANAGERASSISTANT PRODUCTION EDITORCOVER DESIGNERLaurie RosatoneShannon CorlissJonathan CottrellMelissa EdwardsSheena GoldsteinJanis SooYee Lyn SongSeng Ping NgiengThis book was set in 10/12 Times Roman by Thomson Digital, and printed and bound by Hamilton PrintingCompany. The cover was printed by Hamilton Printing Company.Founded in 1807, John Wiley & Sons, Inc. has been a valued source of knowledge and understanding for morethan 200 years, helping people around the world meet their needs and fulfill their aspirations. Our company isbuilt on a foundation of principles that include responsibility to the communities we serve and where we live andwork. In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental,social, economic, and ethical challenges we face in our business. Among the issues we are addressing are carbonimpact, paper specifications and procurement, ethical conduct within our business and among our vendors, andcommunity and charitable support. For more information, please visit our website: www.wiley.com/go/citizenship.1This book is printed on acid-free paper. Copyright # 2011, 2000, 1993, 1983 John Wiley & Sons, Inc. All rights reserved. No part of this publicationmay be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic,mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 ofthe 1976 United States Copyright Act, without either the prior written permission of the Publisher, orauthorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc. 222Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permissionshould be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ07030-5774, (201)748-6011, fax (201)748-6008, website http://www.wiley.com/go/permissions.Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in theircourses during the next academic year. These copies are licensed and may not be sold or transferred to a thirdparty. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions anda free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the UnitedStates, please contact your local representative.Library of Congress Cataloging-in-Publication DataBartle, Robert Gardner, 1927Introduction to real analysis / Robert G. Bartle, Donald R. Sherbert. – 4th ed.p. cm.Includes index.ISBN 978-0-471-43331-6 (hardback)1. Mathematical analysis. 2. Functions of real variables. I. Sherbert, Donald R., 1935- II. Title.QA300.B294 2011515–dc222010045251Printed in the United States of America10 9 8 7 6 5 4 3 2 1

FDED0112/08/201015:42:42Page 5A TRIBUTEThis edition is dedicated to the memory of Robert G. Bartle, a wonderful friend andcolleague of forty years. It has been an immense honor and pleasure to be Bob’s coauthoron the previous editions of this book. I greatly miss his knowledge, his insights, andespecially his humor.November 20, 2010Urbana, IllinoisDonald R. Sherbert

FDED0212/08/201015:43:51Page 6To Jan, with thanks and love.

FPREF12/09/201014:45:10Page 7PREFACEThe study of real analysis is indispensable for a prospective graduate student of pure orapplied mathematics. It also has great value for any student who wishes to go beyond theroutine manipulations of formulas because it develops the ability to think deductively,analyze mathematical situations and extend ideas to new contexts. Mathematics hasbecome valuable in many areas, including economics and management science as wellas the physical sciences, engineering, and computer science. This book was written toprovide an accessible, reasonably paced treatment of the basic concepts and techniques ofreal analysis for students in these areas. While students will find this book challenging,experience has demonstrated that serious students are fully capable of mastering thematerial.The first three editions were very well received and this edition maintains the samespirit and user-friendly approach as earlier editions. Every section has been examined.Some sections have been revised, new examples and exercises have been added, and a newsection on the Darboux approach to the integral has been added to Chapter 7. There is morematerial than can be covered in a semester and instructors will need to make selections andperhaps use certain topics as honors or extra credit projects.To provide some help for students in analyzing proofs of theorems, there is anappendix on ‘‘Logic and Proofs’’ that discusses topics such as implications, negations,contrapositives, and different types of proofs. However, it is a more useful experience tolearn how to construct proofs by first watching and then doing than by reading abouttechniques of proof.Results and proofs are given at a medium level of generality. For instance, continuousfunctions on closed, bounded intervals are studied in detail, but the proofs can be readilyadapted to a more general situation. This approach is used to advantage in Chapter 11where topological concepts are discussed. There are a large number of examples toillustrate the concepts, and extensive lists of exercises to challenge students and to aid themin understanding the significance of the theorems.Chapter 1 has a brief summary of the notions and notations for sets and functions thatwill be used. A discussion of Mathematical Induction is given, since inductive proofs arisefrequently. There is also a section on finite, countable and infinite sets. This chapter canused to provide some practice in proofs, or covered quickly, or used as background materialand returning later as necessary.Chapter 2 presents the properties of the real number system. The first two sections dealwith Algebraic and Order properties, and the crucial Completeness Property is given inSection 2.3 as the Supremum Property. Its ramifications are discussed throughout theremainder of the chapter.In Chapter 3, a thorough treatment of sequences is given, along with the associatedlimit concepts. The material is of the greatest importance. Students find it rather naturalthough it takes time for them to become accustomed to the use of epsilon. A briefintroduction to Infinite Series is given in Section 3.7, with more advanced materialpresented in Chapter 9.vii

FPREF12/09/201014:45:10viiiPage 8PREFACEChapter 4 on limits of functions and Chapter 5 on continuous functions constitute theheart of the book. The 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 intervalsare discussed in Sections 5.3 and 5.4. The notion of a gauge is introduced in Section 5.5 andused to give alternate proofs of these theorems. Monotone functions are discussed inSection 5.6.The basic theory of the derivative is given in the first part of Chapter 6. This material isstandard, except a result of Carath eodory is used to give simpler proofs of the Chain Ruleand the Inversion Theorem. The remainder of the chapter consists of applications of theMean Value Theorem and may be explored as time permits.In Chapter 7, the Riemann integral is defined in Section 7.1 as a limit of Riemannsums. This has the advantage that it is consistent with the students’ first exposure to theintegral in calculus, and since it is not dependent on order properties, it permits immediategeneralization to complex- and vector-values functions that students may encounter in latercourses. It is also consistent with the generalized Riemann integral that is discussed inChapter 10. Sections 7.2 and 7.3 develop properties of the integral and establish theFundamental Theorem of Calculus. The new Section 7.4, added in response to requestsfrom a number of instructors, develops the Darboux approach to the integral in terms ofupper and lower integrals, and the connection between the two definitions of the integral isestablished. Section 7.5 gives a brief discussion of numerical methods of calculating theintegral of continuous functions.Sequences of functions and uniform convergence are discussed in the first two sectionsof Chapter 8, and the basic transcendental functions are put on a firm foundation inSections 8.3 and 8.4. Chapter 9 completes the discussion of infinite series that was begunin Section 3.7. Chapters 8 and 9 are intrinsically important, and they also show how thematerial in the earlier chapters can be applied.Chapter 10 is a presentation of the generalized Riemann integral (sometimes called the‘‘Henstock-Kurzweil’’ or the ‘‘gauge’’ integral). It will be new to many readers and theywill be amazed that such an apparently minor modification of the definition of the Riemannintegral can lead to an integral that is more general than the Lebesgue integral. Thisrelatively new approach to integration theory is both accessible and exciting to anyone whohas studied the basic Riemann integral.Chapter 11 deals with topological concepts. Earlier theorems and proofs are extendedto a more abstract setting. For example, the concept of compactness is given properemphasis and metric spaces are introduced. This chapter will be useful to studentscontinuing on to graduate courses in mathematics.There are lengthy lists of exercises, some easy and some challenging, and ‘‘hints’’ tomany of them are provided to help students get started or to check their answers. Morecomplete solutions of almost every exercise are given in a separate Instructor’s Manual,which is available to teachers upon request to the publisher.It is a satisfying experience to see how the mathematical maturity of the studentsincreases as they gradually learn to work comfortably with concepts that initially seemedso mysterious. But there is no doubt that a lot of hard work is required on the part of both thestudents and the teachers.Brief biographical sketches of some famous mathematicians are included to enrich thehistorical perspective of the book. Thanks go to Dr. Patrick Muldowney for his photographof Professors Henstock and Kurzweil, and to John Wiley & Sons for obtaining portraits ofthe other mathematicians.

FPREF12/09/201014:45:10Page 9PREFACEixMany helpful comments have been received from colleagues who have taught fromearlier editions of this book and their remarks and suggestions have been appreciated. Iwish to thank them and express the hope that they find this new edition even more helpfulthan the earlier ones.November 20, 2010Urbana, IllinoisDonald R. SherbertTHE GREEK iOmega

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FTOC12/08/201015:45:54Page 11CONTENTSCHAPTER 1 PRELIMINARIES 11.11.21.3Sets and Functions 1Mathematical Induction 12Finite and Infinite Sets 16CHAPTER 2 THE REAL NUMBERS 232.12.22.32.42.5The Algebraic and Order Properties of R 23Absolute Value and the Real Line 32The Completeness Property of R 36Applications of the Supremum Property 40Intervals 46CHAPTER 3 SEQUENCES AND SERIES 543.13.23.33.43.53.63.7Sequences and Their Limits 55Limit Theorems 63Monotone Sequences 70Subsequences and the Bolzano-Weierstrass Theorem 78The Cauchy Criterion 85Properly Divergent Sequences 91Introduction to Infinite Series 94CHAPTER 4 LIMITS 1024.14.24.3Limits of Functions 103Limit Theorems 111Some Extensions of the Limit Concept 116CHAPTER 5 CONTINUOUS FUNCTIONS 1245.15.25.35.45.55.6Continuous Functions 125Combinations of Continuous Functions 130Continuous Functions on Intervals 134Uniform Continuity 141Continuity and Gauges 149Monotone and Inverse Functions 153xi

FTOC12/08/201015:45:54xiiPage 12CONTENTSCHAPTER 6 DIFFERENTIATION 1616.16.26.36.4The Derivative 162The Mean Value Theorem 172L’Hospital’s Rules 180Taylor’s Theorem 188CHAPTER 7 THE RIEMANN INTEGRAL 1987.17.27.37.47.5Riemann Integral 199Riemann Integrable Functions 208The Fundamental Theorem 216The Darboux Integral 225Approximate Integration 233CHAPTER 8 SEQUENCES OF FUNCTIONS 2418.18.28.38.4Pointwise and Uniform Convergence 241Interchange of Limits 247The Exponential and Logarithmic Functions 253The Trigonometric Functions 260CHAPTER 9 INFINITE SERIES 2679.19.29.39.4Absolute Convergence 267Tests for Absolute Convergence 270Tests for Nonabsolute Convergence 277Series of Functions 281CHAPTER 10 THE GENERALIZED RIEMANN INTEGRAL 28810.110.210.310.4Definition and Main Properties 289Improper and Lebesgue Integrals 302Infinite Intervals 308Convergence Theorems 315CHAPTER 11 A GLIMPSE INTO TOPOLOGY 32611.111.211.311.4Open and Closed Sets in R 326Compact Sets 333Continuous Functions 337Metric Spaces 341APPENDIX A LOGIC AND PROOFS 348APPENDIX B FINITE AND COUNTABLE SETS 357

FTOC12/08/201015:45:54Page 13CONTENTSxiiiAPPENDIX C THE RIEMANN AND LEBESGUE CRITERIA 360APPENDIX D APPROXIMATE INTEGRATION 364APPENDIX E TWO EXAMPLES 367REFERENCES 370PHOTO CREDITS 371HINTS FOR SELECTED EXERCISES 372INDEX 395

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C0112/08/201012:3:14Page 1CHAPTER 1PRELIMINARIESIn this initial chapter we will present the background needed for the study of realanalysis. Section 1.1 consists of a brief survey of set operations and functions, two vitaltools for all of mathematics. In it we establish the notation and state the basicdefinitions and properties that will be used throughout the book. We will regard theword ‘‘set’’ as synonymous with the words ‘‘class,’’ ‘‘collection,’’ and ‘‘family,’’ andwe will not define these terms or give a list of axioms for set theory. This approach,often referred to as ‘‘naive’’ set theory, is quite adequate for working with sets in thecontext of real analysis.Section 1.2 is concerned with a special method of proof called MathematicalInduction. It is related to the fundamental properties of the natural number system and,though it is restricted to proving particular types of statements, it is important and usedfrequently. An informal discussion of the different types of proofs that are used inmathematics, such as contrapositives and proofs by contradiction, can be found inAppendix 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 is astarting point.Section 1.1 Sets and FunctionsTo the reader: In this section we give a brief review of the terminology and notation thatwill be used in this text. We suggest that you look through it quickly and come back laterwhen you need to recall the meaning of a term or a symbol.If an element x is in a set A, we writex2Aand say that x is a member of A, or that x belongs to A. If x is not in A, we writex2 A:If every element of a set A also belongs to a set B, we say that A is a subset of B and writeA Bor B A:1

C0112/08/201012:3:152Page 2CHAPTER 1 PRELIMINARIESWe say that a set A is a proper subset of a set B if A B, but there is at least one element ofB that is not in A. In this case we sometimes writeA B:1.1.1 Definition Two sets A and B are said to be equal, and we write A ¼ B, if theycontain the same elements.Thus, to prove that the sets A and B are equal, we must show thatA BandB 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 writefx 2 S : PðxÞgfor 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. (We will use the symbol :¼ to mean that the symbol on the left is being definedby the symbol on the right.) TheThe The The setsetsetsetofofofofnatural numbers N :¼ f1; 2; 3; . . .g,integers Z :¼ f0; 1; 1; 2; 2; . . .g,rational numbers Q :¼ fm n : m; n 2 Z and n 6¼ 0g,real numbers R.The set R of real numbers is of fundamental importance and will be discussed at lengthin Chapter 2.1.1.2 Examples (a) The set x 2 N : x2 3x þ 2 ¼ 0 consists of those natural numbers satisfying the stated equation. Since the only solutions ofthis quadratic equation are x ¼ 1 and x ¼ 2, we can denote this set more simply by {1, 2}.(b) A natural number n is even if it has the form n ¼ 2k for some k 2 N. The set of evennatural numbers can be writtenf2k : k 2 N g;which is less cumbersome than fn 2 N : n ¼ 2k; k 2 N g. Similarly, the set of odd naturalnumbers can be writtenf2k 1 : k 2 N g:&Set 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, itis 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 ‘‘and/or.’’

C0112/08/201012:3:15Page 31.1 SETS AND FUNCTIONS31.1.3 Definition (a) The union of sets A and B is the setA [ B :¼ fx : x 2 A or x 2 Bg:(b) The intersection of the sets A and B is the setA \ B :¼ fx : x 2 A and x 2 Bg:(c) The complement of B relative to A is the setAnB :¼ fx : x 2 A and x 2 Bg:Figure 1.1.1(a) A [ B(b) A \ B(c) AnBThe set that has no elements is called the empty set and is denoted by the symbol ;.Two sets A and B are said to be disjoint if they have no elements in common; this can beexpressed by writing A \ B ¼ ;.To illustrate the method of proving set equalities, we will next establish one of theDe Morgan laws for three sets. The proof of the other one is left as an exercise.1.1.4 Theorem If A, B, C are sets, then(a) AnðB [ C Þ ¼ ðAnBÞ \ ðAnC Þ,(b) AnðB \ C Þ ¼ ðAnBÞ [ ðAnC Þ.Proof. To prove (a), we will show that every element in AnðB [ C Þ is contained in both(AnB) and (AnC), and conversely.If x is in AnðB [ CÞ, then x is in A, but x is not in B [ C. Hence x is in A, but x is neitherin B nor in C. Therefore, x is in A but not B, and x is in A but not C. Thus, x 2 AnB andx 2 AnC, which shows that x 2 ðAnBÞ \ ðAnC Þ.Conversely, if x 2 ðAnBÞ \ ðAnCÞ, then x 2 ðAnBÞ and x 2 ðAnCÞ. Hence x 2 A andboth x 2 B and x 2 C. Therefore, x 2 A and x 2 ðB [ CÞ, so that x 2 AnðB [ C Þ.Since the sets ðAnBÞ \ ðAnC Þ and AnðB [ C Þ contain the same elements, they areequal by Definition 1.1.1.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 {A1, A2, . . . , An}, their union is the set A consisting ofall elements that belong to at least one of the sets Ak, and their intersection consists of allelements that belong to a

Introduction to real analysis / Robert G. Bartle, Donald R. Sherbert. – 4th ed. p. cm. Includes index. ISBN 978-0-471-43331-6 (hardback) 1. Mathematical analysis. 2. Functions of real variables. I. Sherbert, Donald R., 1935- II. Title. QA300.B294 2011 515–dc22 2010045251 Printed in the United States of America 10987654321. FDED01 12/08/2010 15:42:42 Page 5 A TRIBUTE This edition is .

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