First Course in Real Analysis PDF
Preview First Course in Real Analysis
Undergraduate Texts in Mathematics Editors S. Axler F. W. Gehring K.A. Ribet Springer Science+Business Media, LLC Undergraduate Texts in Mathematics Anglin: Mathematics: A Concise History Devlin: The Joy of Sets: Fundamentals and Philosophy. of Contemporary Set Theory. Readings in Mathematics. Second edition. Anglin/Lambek: The Heritage of Dixmier: General Topology. Thales. Driver: Why Math? Readings in Mathematics. Ebbinghaus/Flumffhomas: Apostol: Introduction to Analytic Mathematical Logic. Second edition. Number Theory. Second edition. Edgar: Measure, Topology, and Fractal Armstrong: Basic Topology. Geometry. Armstrong: Groups and Symmetry. Elaydi: Introduction to Difference Axler: Linear Algebra Done Right. Equations. Second edition. Exner: An Accompaniment to Higher Beardon: Limits: A New Approach to Mathematics. Real Analysis. FineJRosenberger: The Fundamental BakINewman: Complex Analysis. Theory of Algebra. Second edition. Fischer: Intermediate Real Analysis. BanchoffIWermer: Linear Algebra Flanigan/Kazdan: Calculus Two: Linear Through Geometry. Second edition. and Nonlinear Functions. Second Berberian: A First Course in Real edition. Analysis. Fleming: Functions of Several Variables. Bremaud: An Introduction to Second edition. Probabilistic Modeling. Foulds: Combinatorial Optimization for Bressoud: Factorization and Primality Undergraduates. Testing. Foulds: Optimization Techniques: An Bressoud: Second Year Calculus. Introduction. Readings in Mathematics. Franklin: Methods of Mathematical Brickman: Mathematical Introduction Economics. to Linear Programming and Game Gordon: Discrete Probability. Theory. HairerlWanner: Analysis by Its History. Browder: Mathematical Analysis: Readings in Mathematics. An Introduction. Halmos: Finite-Dimensional Vector Buskeslvan Rooij: Topological Spaces: Spaces. Second edition. From Distance to Neighborhood. Halmos: Naive Set Theory. Cederberg: A Course in Modem HiimmerlinIHoffmann: Numerical Geometries. Mathematics. Childs: A Concrete Introduction to Readings in Mathematics. Higher Algebra. Second edition. Hijab: Introduction to Calculus and Chung: Elementary Probability Theory Classical Analysis. with Stochastic Processes. Third HiitonIHoltonlPedersen: Mathematical edition. Reflections: In a Room with Many CoxILittleJO'Shea: Ideals, Varieties, Mirrors. and Algorithms. Second edition. IoosslJoseph: Elementary Stability Croom: Basic Concepts of Algebraic and Bifurcation Theory. Second Topology. edition. Curtis: Linear Algebra: An Introductory Isaac: The Pleasures of Probability. Approach. Fourth edition. Readings in Mathematics. Sterling K. Berberian A First Course in Real Analysis With 19 Illustrations Springer Sterling K. Berberian Department of Mathematics The University of Texas at Austin Austin, TX 78712 USA Editorial Board S. Axler EW. Gehring K.A. Ribet Department of Department of Department of Mathematics Mathematics Mathematics San Francisco State University University of Michigan University of California San Francisco, CA 94132 Ann Arbor, MI 48109 at Berkeley USA USA Berkeley, CA 94720-3840 USA On the cover: Formula for the fundamental theorem of calculus. Mathematics Subject Classification (1991): 26-01 Library of Congress Cataloging-in-Publication Data Berberian, Sterling K., 1926- A first course in real analysis/Sterling K. Berberian p. cm. - (Undergraduate texts in mathematics) lncIudes bibliographical references (p. - ) and index. ISBN 978-1-4612-6433-0 ISBN 978-1-4419-8548-4 (eBook) DOI 10.1007/978-1-4419-8548-4 1. Mathematical analysis. 2. Numbers, Real. I. Title. II. Series. QA300.B457 1994 S15-dc20 93-46020 Printed on acid-free paper. © 1994 Springer Science+Business Media New York Origina11y published by Springer-Verlag New York, lne. in 1994 Softcover reprint of the hardcover Ist edition 1994 Ali rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter deve10ped is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely byanyone. Production managed by Francine McNei11; manufacturing supervised by Genieve Shaw. Photocomposed copy prepared using the author's TeX files. 9 8 7 6 5 4 3 2 (Corrected seeond printing, 1998) ISBN 978-1-4612-6433-0 SPIN 10662888 For Carol Preface Mathematics is the music of science, and real analysis is the Bach of mathematics. There are many other foolish things I could say about the subject of this book, but the foregoing will give the reader an idea of where my heart lies. The present book was written to support a first course in real analysis, normally taken after a year of elementary calculus. Real analysis is, roughly speaking, the modern setting for Calculus, "real" alluding to the field of real numbers that underlies it all. At center stage are functions, defined and taking values in sets of real numbers or in sets (the plane, 3-space, etc.) readily derived from the real numbers; a first course in real analysis traditionally places the emphasis on real-valued functions defined on sets of real numbers. The agenda for the course: (1) start with the axioms for the field ofreal numbers, (2) build, in one semester and with appropriate rigor, the foun dations of calculus (including the "Fundamental Theorem"), and, along the way, (3) develop those skills and attitudes that enable us to continue learning mathematics on our own. Three decades of experience with the exercise have not diminished my astonishment that it can be done. This paragraph is about prerequisites. The first three sections of the Appendix summarize the notations and terminology of sets and functions, and a few principles of elementary logic, that are the language in which the subject of the book is expressed. Ideally, the student will have met such matters in a preliminary course (on almost any subject) that provides some experience in the art of theorem-proving. Nonetheless, proofs are written out in sufficient detail that the book can also be used for a "first post-calculus, theorem-proving course"; a couple of days of orientation on material in the Appendix will be needed, and the going will be slower, at least at the beginning, but it can be done. Some topics will have to be sacrificed to accommodate the slower pace; I recommend sacrificing all of Chapters 10 (Infinite series) and 11 (Beyond the Riemann integral), the last four sections of Chapter 9, and Theorem 8.5.5 (used only in one of the sections of Chapter 9 just nominated for omission). Chapter 11 is an attempt to bring some significant part of the theory of the Lebesgue integral-a powerful generalization of the integral of elemen tary calculus-within reach of an undergraduate course in analysis, while vii viii Preface avoiding the sea of technical details that usually accompanies it. It will not be easy to squeeze this chapter into a one-semester course- perhaps it should not be tried (I never did in my classes)- but it is good to have a glimpse of what lies over the horizon. Probably the best use for this chapter is a self-study project for the student headed for a graduate course in real analysis; just reading the chapter introduction should be of some help in navigating the above-mentioned sea of details. Each section is followed by an exercise set. The division of labor between text and exercises is straightforward: no proof in the text depends on the solution of an exercise and, throughout Chapters 1-10, the exercises are meant to be solvable using concepts and theorems already covered in the text (the very few exceptions to this rule are clearly signaled). On the other hand, many of the 'exercises' in Chapter 11 are included primarily for the reader's information and do require results not covered in the text; where this is the case, references to other sources are provided. Some exercises contain extensions and generalizations (for example, the Riemann-Stieltjes integral and metric space topology) of material in the text proper; an exceptionally well-prepared class can easily integrate such topics into the mainstream. Other exercises offer alternative proofs of the orems proved in the text. Typically, exercises are for 'hands-on' practice, to strengthen the reader's grasp of the concepts introduced in that section; the easiest are good candidates for inclusion on a test, which is how many of them came into being. Experience persuades me that sequential convergence is the most natural way to introduce limits. This form of the limit concept has much intu itive support (decimal expansions, the half-the-distance-to-the-wall story, etc.), but 1 acknowledge that the decision is subjective and that 1 have not always taught the course that way. The equivalent epsilon-delta and 'neigh borhood' formulations of limits (superior to sequences in some situations) are also discussed fully. My observation is that the introductory course in analysis, more than any other in the undergraduate mathematics curriculum, is a course that makes a difference. The right course at the right time, distilling millennia of human reflection into a brief semester's time, it is a wonderful learning machine whose output is, no less, Mathematicians. Austin, Texas Sterling Berberian Summer, 1993 Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER 1 Axioms for the Field IR of Real Numbers 1 § 1.1. The field axioms . . . . . 1 §1.2. The order axioms . . . . . . . . . . . 4 §1.3. Bounded sets, LUB and GLB . . . . . . 8 §1.4. The completeness axiom (existence of LUB's) .11 CHAPTER 2 First Properties of IR . . . . . . . . . . . . . . . . . 15 §2.1. Dual of the completeness axiom (existence of GLB's) 15 §2.2. Archimedean property . · 16 §2.3. Bracket function · 19 §2.4. Density of the rationals · 20 §2.5. Monotone sequences . . · 21 §2.6. Theorem on nested intervals · 23 §2.7. Dedekind cut property . 26 §2.8. Square roots . · 28 §2.9. Absolute value . . . .30 CHAPTER 3 Sequences of Real Numbers, Convergence .. 33 §3.1. Bounded sequences · 33 §3.2. Ultimately, frequently · 35 §3.3. Null sequences .36 §3.4. Convergent sequences · 39 §3.5. Subsequences, Weierstrass-Bolzano theorem · 43 §3.6. Cauchy's criterion for convergence. . . .48 §3.7. limsup and liminf of a bounded sequence . . · 50 ix x Contents CHAPTER 4 Special Subsets of lR. ... 56 §4.1. Intervals . . . .56 §4.2. Closed sets . . .61 §4.3. Open sets, neighborhoods .65 §4.4. Finite and infinite sets . . .71 §4.5. Heine-Borel covering theorem . 75 CHAPTER 5 Continuity . . . . . . . . . . . . . . .. 80 §5.1. Functions, direct images, inverse images .80 §5.2. Continuity at a point .84 §5.3. Algebra of continuity .88 §5.4. Continuous functions .90 §5.5. One-sided continuity. . 92 §5.6. Composition . . . . .96 CHAPTER 6 Continuous Functions on an Interval .. 98 §6.1. Intermediate value theorem . . . .98 §6.2. n'th roots 100 §6.3. Continuous functions on a closed interval . 101 §6.4. Monotonic continuous functions 103 §6.5. Inverse function theorem 105 §6.6. Uniform continuity . . . . . 106 CHAPTER 7 Limits of Functions 110 §7.1. Deleted neighborhoods 110 §7.2. Limits . . . . . . . 112 §7.3. Limits and continuity 115 §7.4. E,8 characterization of limits 117 §7.5. Algebra of limits . . . . . 118 CHAPTER 8 Derivatives . 121 §8.1. Differentiability . . . 121 §8.2. Algebra of derivatives 124 §8.3. Composition (Chain Rule) 126 §8.4. Local max and min 128 §8.5. Mean value theorem . . . 129
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