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INTRODUCTIONTO REAL ANALYSIS

William F. Trench

Professor Emeritus

Trinity University

San Antonio, TX, USA

©2003 William F. Trench, all rights reserved

Library of Congress Cataloging-in-PublicationData

Trench, William F.

Introduction to real analysis / William F. Trench

p. cm.

ISBN 0-13-045786-8

1. Mathematical Analysis. I. Title.

QA300.T6672003

515-dc21 2002032369

Free Edition 1, March 2009

This book was published previously by Pearson Education. This free edition is made available in the hope that it will beuseful as a textbook or refer- ence. Reproduction is permitted for any valid noncommercial educational, mathematical, or scientific purpose. It may be posted on faculty web pages for convenience of student downloads. However, sale or charges for profit beyond reasonable printing costs are pro- hibited. A complete instructor"ssolutionmanual is available by email to wtrench@trinity.edu, sub- ject to verification of the requestor"s faculty status.

TO BEVERLY

Contents

Prefacevi

Chapter 1 The Real Numbers1

1.1 The Real Number System1

1.2 Mathematical Induction10

1.3 The Real Line19

Chapter 2 Differential Calculus of Functions of One Variable30

2.1 Functions and Limits30

2.2 Continuity53

2.3 Differentiable Functions of One Variable 73

2.4 L"Hospital"s Rule88

2.5 Taylor"s Theorem98

Chapter 3 Integral Calculus of Functions of One Variable 113

3.1 Definition of the Integral 113

3.2 Existence of the Integral128

3.3 Properties of the Integral 135

3.4 Improper Integrals151

3.5 A More Advanced Look at the Existence

of the Proper Riemann Integral 171

Chapter 4 Infinite Sequences and Series 178

4.1 Sequences of Real Numbers 179

4.2 Earlier Topics Revisited With Sequences 195

iv

Contentsv

4.3 Infinite Series of Constants 200

4.4 Sequences and Series of Functions 234

4.5 Power Series257

Chapter 5 Real-Valued Functions of Several Variables 281

5.1 Structure ofRRRn281

5.2 Continuous Real-Valued Function ofnVariables 302

5.3 Partial Derivatives and the Differential 316

5.4 The Chain Rule and Taylor"s Theorem 339

Chapter 6 Vector-Valued Functions of Several Variables 361

6.1 Linear Transformations and Matrices 361

6.2 Continuity and Differentiability of Transformations 378

6.3 The Inverse Function Theorem 394

6.4 The Implicit Function Theorem 417

Chapter 7 Integrals of Functions of Several Variables 435

7.1 Definition and Existence of the Multiple Integral 435

7.2 Iterated Integrals and Multiple Integrals 462

7.3 Change of Variables in Multiple Integrals 484

Chapter 8 Metric Spaces518

8.1 Introduction to Metric Spaces 518

8.2 Compact Sets in a Metric Space 535

8.3 Continuous Functions on Metric Spaces 543

Answers to Selected Exercises 549

Index563

Preface

This is a text for a two-term course in introductoryreal analysis for junior or senior math- ematics majors and science students with a serious interestin mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathe- matical maturity that can be gained from an introductoryreal analysis course. The book is designed to fill the gaps left in the development ofcalculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calcu- lus sequence is the onlyspecific prerequisitefor Chapters 1-5, which deal withreal-valued functions. (However, other analysis oriented courses, such as elementary differential equa- tion, also provide useful preparatory experience.) Chapters 6 and 7 require a working knowledge of determinants, matrices and linear transformations, typically available from a first course in linear algebra. Chapter 8 is accessible aftercompletion of Chapters 1-5. Without taking a position for or against the current reformsin mathematics teaching, I think it is fair to say that the transition from elementary courses such as calculus, linear algebra, and differential equations to a rigorous real analysis course is a bigger step to- day than it was just a few years ago. To make this step today"s students need more help than their predecessors did, and must be coached and encouraged more. Therefore, while strivingthroughout to maintain a high level of rigor, I havetried to write as clearly and in- formally as possible. In this connection I find it useful to address the student in the second person. I have included 295 completely worked out examples to illustrate and clarify all major theorems and definitions. I have emphasized careful statements of definitions and theorems and have tried to be complete and detailed in proofs, except for omissions left to exercises. I give a thorough treatment of real-valued functions before considering vector-valued functions. In making the transitionfrom onetoseveral variables and fromreal-valuedto vector-valuedfunctions, I have left to the student some proofs that are essentially repetitions of earlier theorems. I believe that workingthroughthe detailsof straightforwardgeneralizations ofmore elemen- tary results is good practice for the student. vi

Prefacevii

Great care has gone into the preparation of the 760 numbered exercises, many with multiple parts. They range from routine to very difficult. Hints are provided for the more difficult parts of the exercises.

Organization

Chapter 1 is concerned with the real number system. Section 1.1 begins with a brief dis- cussion of the axioms for a complete ordered field, but no attempt is made to develop the reals from them; rather, it is assumed that the student is familiar with the consequences of these axioms, except for one: completeness. Since the difference between a rigorous and nonrigorous treatment of calculus can be described largelyin terms of the attitude taken toward completeness, I have devoted considerable effort todeveloping its consequences. Section 1.2 is about induction. Although this may seem out ofplace in a real analysis course, I have found that the typical beginning real analysis student simply cannot do an inductionproofwithoutreviewingthemethod. Section1.3is devotedtoelementary set the- ory and the topologyof the real line, ending withthe Heine-Borel and Bolzano-Weierstrass theorems. Chapter 2 covers the differential calculus of functions of one variable: limits, continu- ity, differentiablility,L"Hospital"s rule, and Taylor"stheorem. The emphasis is on rigorous presentation of principles; no attempt is made to develop the properties of specific ele- mentary functions. Even though this may not be done rigorously in most contemporary calculus courses, I believe that the student"s time is better spent on principles rather than on reestablishing familiar formulas and relationships. Chapter 3 is to devoted to the Riemann integral of functions of one variable. In Sec- tion 3.1 the integral is defined in the standard way in terms ofRiemann sums. Upper and lower integrals are also defined there and used in Section 3.2to study the existence of the integral. Section 3.3 is devoted to properties of the integral. Improper integrals are studied in Section 3.4. I believe that my treatment of improper integrals is more detailed than in most comparable textbooks. A more advanced lookat the existence of the proper Riemann integral is given in Section 3.5, which concludes with Lebesgue"s existence criterion. This section can be omitted without compromising the student"s preparedness for subsequent sections. Chapter 4 treats sequences and series. Sequences of constant are discussed in Sec- tion 4.1. I have chosen to make the concepts of limit inferiorand limit superior parts of this development, mainly because this permits greater flexibility and generality, with little extra effort, in the study of infinite series. Section4.2 provides a brief introduction to the way in which continuityand differentiabilitycan be studied by means of sequences. Sections 4.3-4.5 treat infinite series of constant, sequences and infinite series of functions, and power series, again in greater detail than in most comparable textbooks. The instruc- tor who chooses not to cover these sections completely can omit the less standard topics without loss in subsequent sections. Chapter 5 is devoted to real-valued functions of several variables. It begins with a dis- cussion of the toplogy ofRnin Section 5.1. Continuityand differentiabilityare discussed in Sections 5.2 and 5.3. The chain rule and Taylor"s theorem are discussed in Section 5.4. viiiPreface Chapter6 covers the differentialcalculus of vector-valued functionsof several variables. Section 6.1 reviews matrices, determinants, and linear transformations, which are integral parts of the differential calculus as presented here. In Section 6.2 the differential of a vector-valued functionis defined as a linear transformation, and the chain rule is discussed in terms of composition of such functions. The inverse function theorem is the subject of Section 6.3, where the notion of branches of an inverse is introduced. In Section 6.4 the implicit function theorem is motivated by first consideringlinear transformations and then stated and proved in general. Chapter 7 covers the integral calculus of real-valued functionsof several variables. Mul- tiple integrals are defined in Section 7.1, first over rectangular parallelepipeds and then over more general sets. The discussion deals withthe multipleintegral of a functionwhose discontinuities form a set of Jordan content zero. Section 7.2 deals with the evaluation by iterated integrals. Section 7.3 begins with the definition of Jordan measurability, followed by a derivation of the rule for change of content under a linear transformation, an intuitive formulation of the rule for change of variables in multiple integrals, and finally a careful statement and proof of the rule. The proof is complicated, but this is unavoidable. Chapter 8 deals with metric spaces. The concept and properties of a metric space are introduced in Section 8.1. Section 8.2 discusses compactness in a metric space, and Sec- tion 8.3 discusses continuous functions on metric spaces. Although this book has been published previously in hard copy, this electronic edition should be regarded as a first edition, since producing it involved the nontrivial task of combining L ATEX files that were originallysubmitted to the publisher separately, and intro- ducingnew fonts. Hence, there are undoubtedlyerrors-mathematical and typographical-in this edition. Corrections are welcome and will be incorporated when received.

William F. Trench

wtrench@trinity.edu

Home: 659 HopkintonRoad

Hopkinton,NH 03229

CHAPTER 1

The Real Numbers

INTHIS CHAPTER we begin thestudyofthereal number system. The concepts discussed here will be used throughoutthe book. SECTION 1.1 deals with the axioms that define the real numbers, definitions based on them, and some basic properties that follow from them. SECTION 1.2 emphasizes the principle of mathematical induction. SECTION 1.3 introduces basic ideas of set theory in the context of sets of real num- bers. In this section we prove two fundamental theorems: theHeine-Borel and Bolzano-

Weierstrass theorems.

1.1 THE REAL NUMBER SYSTEM

Having taken calculus, you know a lot about the real number system; however, you probably do not know that all its properties follow from a fewbasic ones. Although we will not carry out the development of the real number system from these basic properties, it is useful to state them as a starting point for the study of real analysis and also to focus on one property, completeness, that is probably new to you.

Field Properties

The real number system (which we will often call simply thereals) is first of all a set fa;b;c;:::gon which the operations of addition and multiplication are defined so thatquotesdbs_dbs4.pdfusesText_7

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