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Introduction to Real Analysis John D. DePree Charles W. Introduction to Real Analysis Introduction to Real Analysis John DePree New Mexico State University Charles Swartz New Mexico State University WILEY John Wiley & Sons New York• Chichester• Brisbane •Toronto• Singapore Copyright © 1988, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Sectons 107 and 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons. Library of Congress Catalogingin Publication Daiw Depree, John D., 1933— Introduction to real analysis. Bibliography: p. 347 Includesindex. 1. Functions of real variables. 2. Mathematical analysis. I. Swartz, Charles, 1938— II. Title. . QA331.5.D46 1988 515.8 87-34622 ISBN 0-471-85391-7 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Preface Traditionally there has been a separation of advanced undergraduate and beginning graduate real analysis into two courses, "Advanced Calculus" and "Introduction to Real Analysis," according to the presumed needs of the students; the former was taken by students in engineering and other applied areas and the latter was taken by students whose primary interest lies in the mathematical sciences. At our institution, these courses have coalesced into a course called "Introduction to Real Analysis," and it contains substantial portions of each of its predecessors. The reasons for this merger are partly to effect an economy of scale, partly because of the increased sophistication of the needs of those interested in the application and use of mathematics in their own primary areas, and partly because of the insufficient mathematical prepa- ration of many students. This text evolved from a set of lecture notes that was written for this new course. We try to begin gently by reviewing and extending some familiar ideas such as the set of real numbers and the convergence of sequences and series of real numbers. The student should have gained familiarity with this introductory material during his or her presumed work in the calculus. The material is very V vi Preface computational in nature and offers an opportunity to become acquainted with the level of rigor that will be demanded. Euclidean space is introduced early, and analysis in this n-dimensional space is stressed throughout the first third of the text. The study of continuity and differentiation in R'1 is carried out with minimal frills by using the sequential methods introduced earlier; the emphasis is on describing the main results quickly, thoroughly, and efficiently. This approach should benefit students in disciplines such as statistics, oper- ations research, and engineering, for such students often take only one semester of analysis in the hope that it will adequately provide them with the background necessary to understand the mathematical analysis encountered in these applications. A particular aid to these students should be the section on optimization in R which includes the Kuhn—Tucker theorem on Lagrange multipliers. The middle third of the text gives a unified treatment of integration theory through the introduction of the gauge integral. Because of its simplicity and ease of exposition, the Riemann integral is the vehicle by which most students of analysis are introduced to integration theory. As deeper applications are encountered, more sophisticated theories of integration are required, and soon there is a variety of integrals, for example, Riemann, Stieltjes, Riemann- Stieltjes, Daniell, and Lebesgue, each requiring its own foundation and struc- ture. Perhaps the most important of these are the integrals of Riemann and Lebesgue. It is recognized that the Lebesgue integral is much superior to the Riemann integral, but because of the technical difficulties encountered in defining and developing the Lebesgue integral, it is often considered inap- propriate for a beginning real analysis course. The gauge integral, introduced in Chapter 13, seems to be an almost perfect combination of the desired properties of the Riemann and Lebesgue integrals. It is conceptually almost as easy to describe as the Riemann integral, and it has all the powerful convergence properties of the Lebesgue integral. The gauge integral is studied in detail; applications and examples are given to illustrate the utility and power of the integral. There has been no attempt at achieving maximum generality. The exposition presents the most important results in a form that should be sufficient for most applications encountered by the student. The last third of the text introduces and studies metric spaces in detail. In this context, a more detailed study of topological notions is made. By this time, the students should have reached a level of sophistication at which these abstract results can be understood and appreciated. The previous work on convergence, continuity, differentiation, and integration in R provides a wealth of examples of metric spaces. Applications of the abstract ideas that are being studied can be given to integral equations, differential equations, and function spaces. The development is in sharp contrast to many introductory texts in which metric spaces are defined early and then examples are given as the exposition develops. For students emerging from a beginning calculus course sequence, the first 18 chapters, with the exception of Chapter 16, when they are covered at a Preface vii moderate pace should constitute a good two-semester course in elementary real analysis. The chapters marked by an asterisk are independent, with the exception that the contraction mapping theorem 23.1 is used in the proof of the inverse function theorem (30.1). These marked chapters can be used to fill out a course according to the needs, interests, and sophistication of the students. We are grateful to those who have helped and encouraged us in this endeavor and particularly to our students who graciously tolerated this work through its many revisions from embryonic class notes to this final version. Special thanks are due to Richard Carmichael, Joe Howard, Joe Kist, Douglas Kurtz, and David Ruch, each of whom worked through the text and offered many valuable suggestions. JOHN DEPREE CHARLES SWARTZ Contents Chapter 1 Preliminaries 1 Chapter 2 Real Numbers 15 Chapter 3 Sequences 23 Chapter 4 Infinite Series 37 Chapter 5 Euclidean Spaces 51 Chapter 6 Limits of Functions 57 ix