ELEMENTARY REAL ANALYSIS ————————————— 2 thomson·bruckner ————————————— Brian S. Thomson Judith B. Bruckner Andrew M. Bruckner www.classicalrealanalysis.com This PDF file is for the text Elementary Real Analysis originally pub- lished by Prentice Hall (Pearson) in 2001. The authors retain the copyright and all commercial uses. [2007] Library of Congress Cataloging-in-Publication Data Thomson,BrianS. Elementaryrealanalysis/BrianS.Thomson, JudithB.Bruckner,AndrewM.Bruckner. p. cm. Includes index. ISBN:0-13-019075-6 1. Mathematical analysis. I.Bruckner,JudithB. II.Bruckner,AndrewM.III.Title QA300.T452001 00-041679 515–dc21 CIP Acquisitions Editor: George Lobell Editor in Chief: Sally Yagan Production Editor: Lynn Savino Wendel Assistant Vice President Production and Manufacturing: David W. Riccardi Executive Managing Editor: Kathleen Schiaparelli Senior Managing Editor: Linda Mihatov Behrens Manufacturing Buyer: Alan Fischer Manufacturing Manager: Trudy Pisciotti Director of Marketing: John Tweeddale Marketing Manager: Angela Battle Marketing Assistant: Vince Jansen Art Director: Jayne Conte Editorial Assistant: Gale Epps Cover Designer: Kiwi Design (cid:1)c Original copyright 2001 by Prentice-Hall, Inc. The authors now retain all rights undercopyright. All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the authors. Originally printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN: 0-13-019075-6 CONTENTS PREFACE xii 1 PROPERTIES OF THE REAL NUMBERS 1 1.1 Introduction 1 1.2 The Real Number System 2 1.3 Algebraic Structure 5 1.4 Order Structure 8 1.5 Bounds 9 1.6 Sups and Infs 10 1.7 The Archimedean Property 13 1.8 Inductive Property of IN 15 1.9 The Rational Numbers Are Dense 16 1.10 The Metric Structure of R 18 1.11 Challenging Problems for Chapter 1 21 2 SEQUENCES 23 2.1 Introduction 23 2.2 Sequences 25 2.2.1 Sequence Examples 26 2.3 Countable Sets 29 2.4 Convergence 32 2.5 Divergence 37 2.6 Boundedness Properties of Limits 39 2.7 Algebra of Limits 41 2.8 Order Properties of Limits 47 2.9 Monotone Convergence Criterion 52 2.10 Examples of Limits 56 2.11 Subsequences 61 2.12 Cauchy Convergence Criterion 65 iii iv 2.13 Upper and Lower Limits 68 2.14 Challenging Problems for Chapter 2 74 3 INFINITE SUMS 77 3.1 Introduction 77 3.2 Finite Sums 78 3.3 Infinite Unordered sums 84 3.3.1 Cauchy Criterion 86 3.4 Ordered Sums: Series 90 3.4.1 Properties 91 3.4.2 Special Series 92 3.5 Criteria for Convergence 98 3.5.1 Boundedness Criterion 99 3.5.2 Cauchy Criterion 99 3.5.3 Absolute Convergence 100 3.6 Tests for Convergence 104 3.6.1 Trivial Test 104 3.6.2 Direct Comparison Tests 105 3.6.3 Limit Comparison Tests 107 3.6.4 Ratio Comparison Test 108 3.6.5 d’Alembert’s Ratio Test 109 3.6.6 Cauchy’s Root Test 111 3.6.7 Cauchy’s Condensation Test 112 3.6.8 Integral Test 114 3.6.9 Kummer’s Tests 115 3.6.10 Raabe’s Ratio Test 118 3.6.11 Gauss’s Ratio Test 118 3.6.12 Alternating Series Test 121 3.6.13 Dirichlet’s Test 122 3.6.14 Abel’s Test 123 3.7 Rearrangements 129 3.7.1 Unconditional Convergence 130 3.7.2 Conditional Con(cid:1)vergence (cid:1) 131 ∞ 3.7.3 Comparison of a and a 133 i=1 i i∈IN i 3.8 Products of Series 135 3.8.1 Products of Absolutely Convergent Series 138 3.8.2 Products of Nonabsolutely Convergent Series 139 3.9 Summability Methods 141 3.9.1 Ces`aro’s Method 142 3.9.2 Abel’s Method 144 3.10 More on Infinite Sums 148 v 3.11 Infinite Products 150 3.12 Challenging Problems for Chapter 3 154 4 SETS OF REAL NUMBERS 158 4.1 Introduction 158 4.2 Points 159 4.2.1 Interior Points 159 4.2.2 Isolated Points 161 4.2.3 Points of Accumulation 161 4.2.4 Boundary Points 162 4.3 Sets 165 4.3.1 Closed Sets 166 4.3.2 Open Sets 167 4.4 Elementary Topology 173 4.5 Compactness Arguments 176 4.5.1 Bolzano-Weierstrass Property 178 4.5.2 Cantor’s Intersection Property 179 4.5.3 Cousin’s Property 181 4.5.4 Heine-Borel Property 182 4.5.5 Compact Sets 186 4.6 Countable Sets 189 4.7 Challenging Problems for Chapter 4 190 5 CONTINUOUS FUNCTIONS 193 5.1 Introduction to Limits 193 5.1.1 Limits (ε-δ Definition) 193 5.1.2 Limits (Sequential Definition) 197 5.1.3 Limits (Mapping Definition) 200 5.1.4 One-Sided Limits 201 5.1.5 Infinite Limits 203 5.2 Properties of Limits 204 5.2.1 Uniqueness of Limits 205 5.2.2 Boundedness of Limits 205 5.2.3 Algebra of Limits 207 5.2.4 Order Properties 210 5.2.5 Composition of Functions 213 5.2.6 Examples 215 5.3 Limits Superior and Inferior 222 5.4 Continuity 223 5.4.1 How to Define Continuity 223 5.4.2 Continuity at a Point 227 vi 5.4.3 Continuity at an Arbitrary Point 230 5.4.4 Continuity on a Set 232 5.5 Properties of Continuous Functions 235 5.6 Uniform Continuity 236 5.7 Extremal Properties 240 5.8 Darboux Property 241 5.9 Points of Discontinuity 243 5.9.1 Types of Discontinuity 243 5.9.2 Monotonic Functions 245 5.9.3 How Many Points of Discontinuity? 249 5.10 Challenging Problems for Chapter 5 251 6 MORE ON CONTINUOUS FUNCTIONS AND SETS 253 6.1 Introduction 253 6.2 Dense Sets 253 6.3 Nowhere Dense Sets 255 6.4 The Baire Category Theorem 257 6.4.1 A Two-Player Game 257 6.4.2 The Baire Category Theorem 259 6.4.3 Uniform Boundedness 260 6.5 Cantor Sets 262 6.5.1 Construction of the Cantor Ternary Set 262 6.5.2 An Arithmetic Construction of K 265 6.5.3 The Cantor Function 267 6.6 Borel Sets 269 6.6.1 Sets of Type G 269 δ 6.6.2 Sets of Type F 271 σ 6.7 Oscillation and Continuity 273 6.7.1 Oscillation of a Function 274 6.7.2 The Set of Continuity Points 277 6.8 Sets of Measure Zero 279 6.9 Challenging Problems for Chapter 6 285 7 DIFFERENTIATION 286 7.1 Introduction 286 7.2 The Derivative 286 7.2.1 Definition of the Derivative 287 7.2.2 Differentiability and Continuity 292 7.2.3 The Derivative as a Magnification 293 7.3 Computations of Derivatives 294 7.3.1 Algebraic Rules 295 vii 7.3.2 The Chain Rule 298 7.3.3 Inverse Functions 302 7.3.4 The Power Rule 303 7.4 Continuity of the Derivative? 305 7.5 Local Extrema 307 7.6 Mean Value Theorem 309 7.6.1 Rolle’s Theorem 310 7.6.2 Mean Value Theorem 312 7.6.3 Cauchy’s Mean Value Theorem 314 7.7 Monotonicity 315 7.8 Dini Derivates 318 7.9 The Darboux Property of the Derivative 322 7.10 Convexity 325 7.11 L’Hoˆpital’s Rule 330 7.11.1 L’Hˆopital’s Rule: 0 Form 332 0 7.11.2 L’Hˆopital’s Rule as x → ∞ 334 ∞ 7.11.3 L’Hˆopital’s Rule: Form 336 ∞ 7.12 Taylor Polynomials 339 7.13 Challenging Problems for Chapter 7 343 8 THE INTEGRAL 346 8.1 Introduction 346 8.2 Cauchy’s First Method 349 8.2.1 Scope of Cauchy’s First Method 351 8.3 Properties of the Integral 354 8.4 Cauchy’s Second Method 359 8.5 Cauchy’s Second Method (Continued) 362 8.6 The Riemann Integral 364 8.6.1 Some Examples 366 8.6.2 Riemann’s Criteria 368 8.6.3 Lebesgue’s Criterion 370 8.6.4 What Functions Are Riemann Integrable? 373 8.7 Properties of the Riemann Integral 374 8.8 The Improper Riemann Integral 378 8.9 More on the Fundamental Theorem of Calculus 380 8.10 Challenging Problems for Chapter 8 382 9 SEQUENCES AND SERIES OF FUNCTIONS 384 9.1 Introduction 384 9.2 Pointwise Limits 385 9.3 Uniform Limits 391 viii 9.3.1 The Cauchy Criterion 394 9.3.2 Weierstrass M-Test 396 9.3.3 Abel’s Test for Uniform Convergence 398 9.4 Uniform Convergence and Continuity 404 9.4.1 Dini’s Theorem 405 9.5 Uniform Convergence and the Integral 408 9.5.1 Sequences of Continuous Functions 408 9.5.2 Sequences of Riemann Integrable Functions 410 9.5.3 Sequences of Improper Integrals 412 9.6 Uniform Convergence and Derivatives 415 9.6.1 Limits of Discontinuous Derivatives 417 9.7 Pompeiu’s Function 419 9.8 Continuity and Pointwise Limits 422 9.9 Challenging Problems for Chapter 9 425 10 POWER SERIES 426 10.1 Introduction 426 10.2 Power Series: Convergence 427 10.3 Uniform Convergence 432 10.4 Functions Represented by Power Series 435 10.4.1 Continuity of Power Series 435 10.4.2 Integration of Power Series 436 10.4.3 Differentiation of Power Series 437 10.4.4 Power Series Representations 440 10.5 The Taylor Series 443 10.5.1 Representing a Function by a Taylor Series 444 10.5.2 Analytic Functions 447 10.6 Products of Power Series 449 10.6.1 Quotients of Power Series 450 10.7 Composition of Power Series 452 10.8 Trigonometric Series 453 10.8.1 Uniform Convergence of Trigonometric Series 454 10.8.2 Fourier Series 455 10.8.3 Convergence of Fourier Series 456 10.8.4 Weierstrass Approximation Theorem 460 11 THE EUCLIDEAN SPACES 462 11.1 The Algebraic Structure of Rn 462 11.2 The Metric Structure of Rn 464 11.3 Elementary Topology of Rn 468 11.4 Sequences in Rn 470 ix 11.5 Functions and Mappings 475 11.5.1 Functions from Rn → R 475 11.5.2 Functions from Rn → Rm 477 11.6 Limits of Functions from Rn → Rm 480 11.6.1 Definition 480 11.6.2 Coordinate-Wise Convergence 483 11.6.3 Algebraic Properties 485 11.7 Continuity of Functions from Rn to Rm 486 11.8 Compact Sets in Rn 489 11.9 Continuous Functions on Compact Sets 490 11.10Additional Remarks 491 12 DIFFERENTIATION ON EUCLIDEAN SPACES 495 12.1 Introduction 495 12.2 Partial and Directional Derivatives 496 12.2.1 Partial Derivatives 497 12.2.2 Directional Derivatives 500 12.2.3 Cross Partials 501 12.3 Integrals Depending on a Parameter 506 12.4 Differentiable Functions 510 12.4.1 Approximation by Linear Functions 511 12.4.2 Definition of Differentiability 512 12.4.3 Differentiability and Continuity 516 12.4.4 Directional Derivatives 517 12.4.5 An Example 519 12.4.6 Sufficient Conditions for Differentiability 521 12.4.7 The Differential 523 12.5 Chain Rules 526 12.5.1 Preliminary Discussion 526 12.5.2 Informal Proof of a Chain Rule 530 12.5.3 Notation of Chain Rules 531 12.5.4 Proofs of Chain Rules (I) 533 12.5.5 Mean Value Theorem 535 12.5.6 Proofs of Chain Rules (II) 536 12.5.7 Higher Derivatives 538 12.6 Implicit Function Theorems 541 12.6.1 One-Variable Case 542 12.6.2 Several-Variable Case 545 12.6.3 Simultaneous Equations 549 12.6.4 Inverse Function Theorem 553 12.7 Functions From R → Rm 556 x 12.8 Functions From Rn → Rm 559 12.8.1 Review of Differentials and Derivatives 560 12.8.2 Definition of the Derivative 562 12.8.3 Jacobians 564 12.8.4 Chain Rules 567 12.8.5 Proof of Chain Rule 569 13 METRIC SPACES 573 13.1 Introduction 573 13.2 Metric Spaces—Specific Examples 575 13.3 Additional Examples 580 13.3.1 Sequence Spaces 580 13.3.2 Function Spaces 582 13.4 Convergence 585 13.5 Sets in a Metric Space 589 13.6 Functions 597 13.6.1 Continuity 599 13.6.2 Homeomorphisms 604 13.6.3 Isometries 610 13.7 Separable Spaces 613 13.8 Complete Spaces 616 13.8.1 Completeness Proofs 617 13.8.2 Subspaces of a Complete Space 619 13.8.3 Cantor Intersection Property 619 13.8.4 Completion of a Metric Space 620 13.9 Contraction Maps 623 13.10Applications of Contraction Maps (I) 630 13.11Applications of Contraction Maps (II) 633 13.11.1Systems of Equations (Example 13.79 Revisited) 634 13.11.2Infinite Systems (Example 13.80 revisited) 635 13.11.3Integral Equations (Example 13.81 revisited) 637 13.11.4Picard’s Theorem (Example 13.82 revisited) 638 13.12Compactness 640 13.12.1The Bolzano-Weierstrass Property 641 13.12.2Continuous Functions on Compact Sets 644 13.12.3The Heine-Borel Property 646 13.12.4Total Boundedness 648 13.12.5Compact Sets in C[a,b] 651 13.12.6Peano’s Theorem 656 13.13Baire Category Theorem 659 13.13.1Nowhere Dense Sets 660