INTRODUCTION TO REAL ANALYSIS - Williams College

1 INTRODUCTIONTO real ANALYSISW illiam F. TrenchProfessor EmeritusTrinity UniversitySan Antonio, TX, USA 2003 William F. Trench, all rights reservedLibrary of Congress Cataloging-in-Publication DataTrench, William to real ANALYSIS / William F. Trenchp. 0-13-045786-81. Mathematical ANALYSIS . I. 2003515-dc212002032369 Free Edition , April 2010 This book was published previously by Pearson 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. However, sale or charges for profit beyond reasonable printing costsare complete instructor s solution manual is available by email to sub-ject to verification of the requestor s faculty BEVERLYC ontentsPrefaceviChapter 1 The real The real Number Mathematical The real Line19 Chapter 2 Differential Calculus of Functions of One Functions and Differentiable Functions of One L Hospital s Taylor s Theorem98 Chapter 3 Integral Calculus of Functions of One Variable Definition of the Existence of the Properties of the Improper A More Advanced Look at the Existenceof the Proper Riemann Integral171 Chapter 4 Infinite Sequences and Sequences of real Earlier Topics Revisited With Infinite Series of Sequences and Series of Power Series257 Chapter 5 real -Valued Functions of Several Structure Continuous real -Valued Function Partial Derivatives and the The Chain Rule

2 And Taylor s Theorem339 Chapter 6 Vector-Valued Functions of Several Variables Linear Transformations and Continuity and Differentiability of The Inverse Function The Implicit Function Theorem417 Chapter 7 Integrals of Functions of Several Definition and Existence of the Multiple Iterated Integrals and Multiple Change of Variables in Multiple Integrals484 Chapter 8 Metric INTRODUCTION to Metric Compact Sets in a Metric Continuous Functions on Metric Spaces543 Answers to Selected Exercises549 Index563 PrefaceThis is a text for a two-term course in introductory real ANALYSIS for junior or senior math-ematics majors and science students with a serious interestin mathematics. Prospectiveeducators or mathematically gifted high school students can also benefit from the mathe-matical maturity that can be gained from an introductory real ANALYSIS book is designed to fill the gaps left in the development ofcalculus as it is usuallypresented in an elementary course, and to provide the background required for insight intomore advanced courses in pure and applied mathematics.

3 The standard elementary calcu-lus sequence is the only specific prerequisite for Chapters 1 5, which deal with real -valuedfunctions. (However, other ANALYSIS oriented courses, such as elementary differential equa-tion, also provide useful preparatory experience.) Chapters 6 and 7 require a workingknowledge of determinants, matrices and linear transformations, typically available from afirst course in linear algebra. Chapter 8 is accessible aftercompletion of Chapters 1 taking a position for or against the current reformsin mathematics teaching, Ithink it is fair to say that the transition from elementary courses such as calculus, linearalgebra, 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 helpthan their predecessors did, and must be coached and encouraged more. Therefore, whilestriving throughout to maintain a high level of rigor, I havetried to write as clearly and in-formally as possible.

4 In this connection I find it useful to address the student in the secondperson. I have included 295 completely worked out examples to illustrate and clarify allmajor theorems and have emphasized careful statements of definitions and theorems and have tried to becomplete and detailed in proofs, except for omissions left to exercises. I give a thoroughtreatment of real -valued functions before considering vector-valued functions. In makingthe transition from one to several variables and from real -valued to vector-valued functions,I have left to the student some proofs that are essentially repetitions of earlier theorems. Ibelieve that working through the details of straightforward generalizations of more elemen-tary results is good practice for the care has gone into the preparation of the 760 numbered exercises, many withmultiple parts. They range from routine to very difficult. Hints are provided for the moredifficult parts of the 1 is concerned with the real number system.

5 Section begins with a brief dis-cussion of the axioms for a complete ordered field, but no attempt is made to develop thereals from them; rather, it is assumed that the student is familiar with the consequences ofthese axioms, except for one: completeness. Since the difference between a rigorous andnonrigorous treatment of calculus can be described largelyin terms of the attitude takentoward completeness, I have devoted considerable effort todeveloping its is about induction. Although this may seem out ofplace in a real analysiscourse, I have found that the typical beginning real ANALYSIS student simply cannot do aninduction proof without reviewing the method. Section is devoted to elementary set the-ory and the topology of the real line, ending with the Heine-Borel and 2 covers the differential calculus of functions of one variable: limits, continu-ity, differentiablility, L Hospital s rule, and Taylor stheorem.

6 The emphasis is on rigorouspresentation 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 contemporarycalculus courses, I believe that the student s time is better spent on principles rather thanon reestablishing familiar formulas and 3 is to devoted to the Riemann integral of functions of one variable. In Sec-tion the integral is defined in the standard way in terms ofRiemann sums. Upper andlower integrals are also defined there and used in Section study the existence of theintegral. Section is devoted to properties of the integral. Improper integrals are studiedin Section I believe that my treatment of improper integrals is more detailed than inmost comparable textbooks. A more advanced look at the existence of the proper Riemannintegral is given in Section , which concludes with Lebesgue s existence criterion.

7 Thissection can be omitted without compromising the student s preparedness for 4 treats sequences and series. Sequences of constant are discussed in Sec-tion I have chosen to make the concepts of limit inferiorand limit superior partsof this development, mainly because this permits greater flexibility and generality, withlittle extra effort, in the study of infinite series. provides a brief introductionto the way in which continuity and differentiability can be studied by means of 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 topicswithout loss in subsequent 5 is devoted to real -valued functions of several variables. It begins with a dis-cussion of the toplogy ofRnin Section Continuity and differentiability are discussedin Sections and The chain rule and Taylor s theorem are discussed in Section 6 covers the differential calculus of vector-valued functions of several reviews matrices, determinants, and linear transformations, which are integralparts of the differential calculus as presented here.

8 In Section the differential of avector-valued function is defined as a linear transformation, and the chain rule is discussedin terms of composition of such functions. The inverse function theorem is the subject ofSection , where the notion of branches of an inverse is introduced. In Section theimplicit function theorem is motivated by first consideringlinear transformations and thenstated and proved in 7 covers the integral calculus of real -valued functions of several variables. Mul-tiple integrals are defined in Section , first over rectangular parallelepipeds and thenover more general sets. The discussion deals with the multiple integral of a function whosediscontinuities form a set of Jordan content zero. Section deals with the evaluation byiterated integrals. Section begins with the definition of Jordan measurability, followedby a derivation of the rule for change of content under a linear transformation, an intuitiveformulation of the rule for change of variables in multiple integrals, and finally a carefulstatement and proof of the rule.

9 The proof is complicated, but this is 8 deals with metric spaces. The concept and properties of a metric space areintroduced in Section Section discusses compactness in a metric space, and Sec-tion discusses continuous functions on metric this book has been published previously in hard copy, this electronic editionshould be regarded as a first edition, since producing it involved the nontrivial task ofcombining LATEX files that were originally submitted to the publisher separately, and intro-ducing new fonts. Hence, there are undoubtedly errors mathematical and typographical inthis edition. Corrections are welcome and will be incorporated when F. 659 Hopkinton RoadHopkinton, NH 03229 CHAPTER 1 The real NumbersIN THIS CHAPTER we begin the study of the real number system. The concepts discussedhere will be used throughout the deals with the axioms that define the real numbers, definitions based onthem, and some basic properties that follow from emphasizes the principle of mathematical introduces basic ideas of set theory in the context of sets of real num-bers.

10 In this section we prove two fundamental theorems: theHeine Borel and Bolzano Weierstrass THE real NUMBER SYSTEMH aving taken calculus, you know a lot about the real number system; however, youprobably do not know that all its properties follow from a fewbasic ones. Although wewill 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 focuson one property, completeness, that is probably new to PropertiesThe real number system (which we will often call simply thereals) is first of all a setfa;b;c;:::gon which the operations of addition and multiplication are defined so thatevery pair of real numbers has a unique sum and product, both real numbers, with thefollowing properties.(A)aCbDbCaandabDba(commutativ e laws).(B). (associative laws).(C) (distributive law).(D)There are distinct real numbers0and1such thataC0 Daanda1 Dafor alla.