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ClassicalRealAnalysis.comELEMENTARYREAL ANALYSISSecond Edition ��Brian S. ThomsonJudith B. BrucknerAndrew M. BrucknerElementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comThis version of Elementary Real Analysis, Second Edition, is a hypertexted pdf file, suitablefor on-screen viewing. For a trade paperback copy of the text, with the same numbering of Theorems andExercises (but with different page numbering), please visit our web site.Direct all correspondence to thomson@sfu.ca.For further information on this title and others in the series visit our website. There are pdf files of thetexts freely available for download as well as instructions on how to order trade paperback copies.www.classicalrealanalysis.comc This second edition is a corrected version of the text Elementary Real Analysis originally published by PrenticeHall (Pearson) in 2001. The authors retain the copyright and all commercial uses.Original Citation: Elementary Real Analysis, Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner.Prentice-Hall, 2001, xv 735 pp. [ISBN 0-13-019075-61]Cover Design and Photography: David SprecherDate PDF file compiled: June 1, 2008Trade Paperback published under ISBN 1-434841-61-8Thomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comCONTENTSPREFACExviiVOLUME ONE11 PROPERTIES OF THE REAL NUMBERS1.1 Introduction1.2 The Real Number System1.3 Algebraic Structure1.4 Order Structure1.5 Bounds1.6 Sups and Infs1.7 The Archimedean Property1.8 Inductive Property of IN1.9 The Rational Numbers Are Dense1.10 The Metric Structure of R1.11 Challenging Problems for Chapter lementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comiv27Notes2 SEQUENCES2.1 Introduction2.2 Sequences2.2.1 Sequence Examples2.3 Countable Sets2.4 Convergence2.5 Divergence2.6 Boundedness Properties of Limits2.7 Algebra of Limits2.8 Order Properties of Limits2.9 Monotone Convergence Criterion2.10 Examples of Limits2.11 Subsequences2.12 Cauchy Convergence Criterion2.13 Upper and Lower Limits2.14 Challenging Problems for Chapter 2Notes29293133374147495260667278848795983 INFINITE SUMS3.1 Introduction3.2 Finite Sums3.3 Infinite Unordered sums3.3.1 Cauchy Criterion3.4 Ordered Sums: Series3.4.1 Properties3.4.2 Special 22123Elementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.com3.53.63.73.83.9vCriteria for Convergence3.5.1 Boundedness Criterion3.5.2 Cauchy Criterion3.5.3 Absolute ConvergenceTests for Convergence3.6.1 Trivial Test3.6.2 Direct Comparison Tests3.6.3 Limit Comparison Tests3.6.4 Ratio Comparison Test3.6.5 d’Alembert’s Ratio Test3.6.6 Cauchy’s Root Test3.6.7 Cauchy’s Condensation Test3.6.8 Integral Test3.6.9 Kummer’s Tests3.6.10 Raabe’s Ratio Test3.6.11 Gauss’s Ratio Test3.6.12 Alternating Series Test3.6.13 Dirichlet’s Test3.6.14 Abel’s TestRearrangements3.7.1 Unconditional Convergence3.7.2 Conditional ConvergencePP3.7.3 Comparison of i 1 ai andi IN aiProducts of Series3.8.1 Products of Absolutely Convergent Series3.8.2 Products of Nonabsolutely Convergent SeriesSummability 7181184186189Elementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comvi3.9.1 Cesàro’s Method3.9.2 Abel’s Method3.10 More on Infinite Sums3.11 Infinite Products3.12 Challenging Problems for Chapter 3Notes1901921972002062114 SETS OF REAL NUMBERS4.1 Introduction4.2 Points4.2.1 Interior Points4.2.2 Isolated Points4.2.3 Points of Accumulation4.2.4 Boundary Points4.3 Sets4.3.1 Closed Sets4.3.2 Open Sets4.4 Elementary Topology4.5 Compactness Arguments4.5.1 Bolzano-Weierstrass Property4.5.2 Cantor’s Intersection Property4.5.3 Cousin’s Property4.5.4 Heine-Borel Property4.5.5 Compact Sets4.6 Countable Sets4.7 Challenging Problems for Chapter y Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comvii5 CONTINUOUS FUNCTIONS5.1 Introduction to Limits5.1.1 Limits (ε-δ Definition)5.1.2 Limits (Sequential Definition)5.1.3 Limits (Mapping Definition)5.1.4 One-Sided Limits5.1.5 Infinite Limits5.2 Properties of Limits5.2.1 Uniqueness of Limits5.2.2 Boundedness of Limits5.2.3 Algebra of Limits5.2.4 Order Properties5.2.5 Composition of Functions5.2.6 Examples5.3 Limits Superior and Inferior5.4 Continuity5.4.1 How to Define Continuity5.4.2 Continuity at a Point5.4.3 Continuity at an Arbitrary Point5.4.4 Continuity on a Set5.5 Properties of Continuous Functions5.6 Uniform Continuity5.7 Extremal Properties5.8 Darboux Property5.9 Points of Discontinuity5.9.1 Types of Discontinuity5.9.2 Monotonic 326328330331333Elementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comviii5.9.3 How Many Points of Discontinuity?5.10 Challenging Problems for Chapter 5Notes3383403426 MORE ON CONTINUOUS FUNCTIONS AND SETS6.1 Introduction6.2 Dense Sets6.3 Nowhere Dense Sets6.4 The Baire Category Theorem6.4.1 A Two-Player Game6.4.2 The Baire Category Theorem6.4.3 Uniform Boundedness6.5 Cantor Sets6.5.1 Construction of the Cantor Ternary Set6.5.2 An Arithmetic Construction of K6.5.3 The Cantor Function6.6 Borel Sets6.6.1 Sets of Type Gδ6.6.2 Sets of Type Fσ6.7 Oscillation and Continuity6.7.1 Oscillation of a Function6.7.2 The Set of Continuity Points6.8 Sets of Measure Zero6.9 Challenging Problems for Chapter 53773783823853923937 DIFFERENTIATION7.1 ary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comix7.2The Derivative7.2.1 Definition of the Derivative7.2.2 Differentiability and Continuity7.2.3 The Derivative as a Magnification7.3 Computations of Derivatives7.3.1 Algebraic Rules7.3.2 The Chain Rule7.3.3 Inverse Functions7.3.4 The Power Rule7.4 Continuity of the Derivative?7.5 Local Extrema7.6 Mean Value Theorem7.6.1 Rolle’s Theorem7.6.2 Mean Value Theorem7.6.3 Cauchy’s Mean Value Theorem7.7 Monotonicity7.8 Dini Derivates7.9 The Darboux Property of the Derivative7.10 Convexity7.11 L’Hôpital’s Rule7.11.1 L’Hôpital’s Rule: 00 Form7.11.2 L’Hôpital’s Rule as x 7.11.3 L’Hôpital’s Rule: Form7.12 Taylor Polynomials7.13 Challenging Problems for Chapter 33435438444448454457460462466471475THE INTEGRALThomson*Bruckner*Bruckner485Elementary Real Analysis, 2nd Edition (2008)

’s First Method8.2.1 Scope of Cauchy’s First Method8.3 Properties of the Integral8.4 Cauchy’s Second Method8.5 Cauchy’s Second Method (Continued)8.6 The Riemann Integral8.6.1 Some Examples8.6.2 Riemann’s Criteria8.6.3 Lebesgue’s Criterion8.6.4 What Functions Are Riemann Integrable?8.7 Properties of the Riemann Integral8.8 The Improper Riemann Integral8.9 More on the Fundamental Theorem of Calculus8.10 Challenging Problems for Chapter 3534VOLUME TWO5369 SEQUENCES AND SERIES OF FUNCTIONS9.1 Introduction9.2 Pointwise Limits9.3 Uniform Limits9.3.1 The Cauchy Criterion9.3.2 Weierstrass M -Test9.3.3 Abel’s Test for Uniform Convergence9.4 Uniform Convergence and Continuity9.4.1 Dini’s *BrucknerElementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comxi9.5Uniform Convergence and the Integral9.5.1 Sequences of Continuous Functions9.5.2 Sequences of Riemann Integrable Functions9.5.3 Sequences of Improper Integrals9.6 Uniform Convergence and Derivatives9.6.1 Limits of Discontinuous Derivatives9.7 Pompeiu’s Function9.8 Continuity and Pointwise Limits9.9 Challenging Problems for Chapter 9Notes56956957157557858058358659059110 POWER SERIES10.1 Introduction10.2 Power Series: Convergence10.3 Uniform Convergence10.4 Functions Represented by Power Series10.4.1 Continuity of Power Series10.4.2 Integration of Power Series10.4.3 Differentiation of Power Series10.4.4 Power Series Representations10.5 The Taylor Series10.5.1 Representing a Function by a Taylor Series10.5.2 Analytic Functions10.6 Products of Power Series10.6.1 Quotients of Power Series10.7 Composition of Power Series10.8 Trigonometric Series10.8.1 Uniform Convergence of Trigonometric 07608612615617620623625628629630Elementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comxii10.8.2 Fourier Series10.8.3 Convergence of Fourier Series10.8.4 Weierstrass Approximation TheoremNotes63163363864011 THE EUCLIDEAN SPACES RN11.1 The Algebraic Structure of Rn11.2 The Metric Structure of Rn11.3 Elementary Topology of Rn11.4 Sequences in Rn11.5 Functions and Mappings11.5.1 Functions from Rn R11.5.2 Functions from Rn Rm11.6 Limits of Functions from Rn Rm11.6.1 Definition11.6.2 Coordinate-Wise Convergence11.6.3 Algebraic Properties11.7 Continuity of Functions from Rn to Rm11.8 Compact Sets in Rn11.9 Continuous Functions on Compact Sets11.10 Additional 667968168268512 DIFFERENTIATION ON RN12.1 Introduction12.2 Partial and Directional Derivatives12.2.1 Partial Derivatives12.2.2 Directional rElementary Real Analysis, 2nd Edition (2008)

iii12.2.3 Cross PartialsIntegrals Depending on a ParameterDifferentiable Functions12.4.1 Approximation by Linear Functions12.4.2 Definition of Differentiability12.4.3 Differentiability and Continuity12.4.4 Directional Derivatives12.4.5 An Example12.4.6 Sufficient Conditions for Differentiability12.4.7 The DifferentialChain Rules12.5.1 Preliminary Discussion12.5.2 Informal Proof of a Chain Rule12.5.3 Notation of Chain Rules12.5.4 Proofs of Chain Rules (I)12.5.5 Mean Value Theorem12.5.6 Proofs of Chain Rules (II)12.5.7 Higher DerivativesImplicit Function Theorems12.6.1 One-Variable Case12.6.2 Several-Variable Case12.6.3 Simultaneous Equations12.6.4 Inverse Function TheoremFunctions From R RmFunctions From Rn Rm12.8.1 Review of Differentials and Derivatives12.8.2 Definition of the 9765768773773777Elementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comxiv12.8.3 Jacobians12.8.4 Chain Rules12.8.5 Proof of Chain RuleNotes77978378679013 METRIC SPACES13.1 Introduction13.2 Metric Spaces—Specific Examples13.3 Additional Examples13.3.1 Sequence Spaces13.3.2 Function Spaces13.4 Convergence13.5 Sets in a Metric Space13.6 Functions13.6.1 Continuity13.6.2 Homeomorphisms13.6.3 Isometries13.7 Separable Spaces13.8 Complete Spaces13.8.1 Completeness Proofs13.8.2 Subspaces of a Complete Space13.8.3 Cantor Intersection Property13.8.4 Completion of a Metric Space13.9 Contraction Maps13.10 Applications of Contraction Maps (I)13.11 Applications of Contraction Maps (II)13.11.1 Systems of Equations (Example 13.79 Revisited)13.11.2 Infinite Systems (Example 13.80 3874Elementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comxv13.11.3 Integral Equations (Example 13.81 revisited)13.11.4 Picard’s Theorem (Example 13.82 revisited)13.12 Compactness13.12.1 The Bolzano-Weierstrass Property13.12.2 Continuous Functions on Compact Sets13.12.3 The Heine-Borel Property13.12.4 Total Boundedness13.12.5 Compact Sets in C[a, b]13.12.6 Peano’s Theorem13.13 Baire Category Theorem13.13.1 Nowhere Dense Sets13.13.2 The Baire Category Theorem13.14 Applications of the Baire Category Theorem13.14.1 Functions Whose Graphs “Cross No Lines”13.14.2 Nowhere Monotonic Functions13.14.3 Continuous Nowhere Differentiable Functions13.14.4 Cantor Sets13.15 Challenging Problems for Chapter 19920921924928VOLUME ONEA-1A APPENDIX: BACKGROUNDA.1 Should I Read This Chapter?A.2 NotationA.2.1 Set NotationA.2.2 Function NotationA.3 What Is nerElementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comxviA.4 Why Proofs?A.5 Indirect ProofA.6 ContrapositionA.7 CounterexamplesA.8 InductionA.9 T INDEXThomson*Bruckner*BrucknerA-28Elementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comPREFACEPreface to the second edition (January 2008)This edition differs from the original 2001 version only in that we corrected a number of misprints andother errors. We are grateful to the many users of that version for notifying us of errors they found. Wewould like to make special mention of Richard Delaware (University of Missouri-Kansas City), and SteveAgronsky (California State Polytechnic University, San Luis Obispo), both of whom went through the entirefirst edition, made many helpful suggestions, and found numerous errors.Original Preface (2001)University mathematics departments have for many years offered courses with titles such as AdvancedCalculus or Introductory Real Analysis. These courses are taken by a variety of students, serve a number ofpurposes, and are written at various levels of sophistication. The students range from ones who have justcompleted a course in elementary calculus to beginning graduate students in mathematics. The purposesare multifold:1. To present familiar concepts from calculus at a more rigorous level.2. To introduce concepts that are not studied in elementary calculus but that are needed in more advancedxviiThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comxviiiPrefaceundergraduate courses. This would include such topics as point set theory, uniform continuity offunctions, and uniform convergence of sequences of functions.3. To provide students with a level of mathematical sophistication that will prepare them for graduatework in mathematical analysis, or for graduate work in several applied fields such as engineering oreconomics.4. To develop many of the topics that the authors feel all students of mathematics should know.There are now many texts that address some or all of these objectives. These books range from onesthat do little more than address objective (1) to ones that try to address all four objectives. The books ofthe first extreme are generally aimed at one-term courses for students with minimal background. Books atthe other extreme often contain substantially more material than can be covered in a one-year course.The level of rigor varies considerably from one book to another, as does the style of presentation. Somebooks endeavor to give a very efficient streamlined development; others try to be more user friendly. Wehave opted for the user-friendly approach. We feel this approach makes the concepts more meaningful tothe student.Our experience with students at various levels has shown that most students have difficulties when topicsthat are entirely new to them first appear. For some students that might occur almost immediately whenrigorous proofs are required, for example, ones needing ε-δ arguments. For others, the difficulties begin withelementary point set theory, compactness arguments, and the like.To help students with the transition from elementary calculus to a more rigorous course, we have includedmotivation for concepts most students have not seen before and provided more details in proofs when weintroduce new methods. In addition, we have tried to give students ample opportunity to see the new toolsin action.For example, students often feel uneasy when they first encounter the various compactness arguments(Heine-Borel theorem, Bolzano-Weierstrass theorem, Cousin’s lemma, introduced in Section 4.5). To helpthe student see why such theorems are useful, we pose the problem of determining circumstances underwhich local boundedness of a function f on a set E implies global boundedness of f on E. We show byexample that some conditions on E are needed, namely that E be closed and bounded, and then show howThomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comPrefacexixeach of several theorems could be used to show that closed and boundedness of the set E suffices. Thus weintroduce students to the theorems by showing how the theorems can be used in natural ways to solve aproblem.We have also included some optional material, marked as “Advanced” or “Enrichment” and flagged withthe symbol .EnrichmentWe have indicated as “Enrichment”‘ some relatively elementary material that could be added to a longercourse to provide enrichment and additional examples. For example, in Chapter 3 we have added to thestudy of series a section on infinite products. While such a topic plays an important role in the representationof analytic functions, it is presented here to allow the instructor to explore ideas that are closely related tothe study of series and that help illustrate and review many of the fundamental ideas that have played arole in the study of series.AdvancedWe have indicated as “Advanced” material of a more mathematically sophisticated nature that can beomitted without loss of continuity. These topics might be needed in more advanced courses in real analysisor in certain of the marked sections or exercises that appear later in this book. For example, in Chapter 2we have added to the study of sequence limits a section on lim sups and lim infs. For an elementary firstcourse this can be considered somewhat advanced and skipped. Later problems and text material thatrequire these concepts are carefully indicated. Thus, even though the text carries on to relatively advancedundergraduate analysis, a first course can be presented by avoiding these advanced sections.We apply these markings to some entire chapters as well as to some sections within chapters and evento certain exercises. We do not view these markings as absolute. They can simply be interpreted in thefollowing ways. Any unmarked material will not depend, in any substantial way, on earlier marked sections.In addition, if a section has been flagged and will be used in a much later section of this book, we indicatewhere it will be required.Thomson*Bruckner*BrucknerElementary Real Analysis, 2nd Edition (2008)

ClassicalRealAnalysis.comxxPrefaceThe material marked “Advanced” is in line with goals (2) and (3). We resist the temptation to addressobjective (4). There are simply too many additional topics that one might feel every student should know(e.g., functions of bounded variation, Riemann-Stieltjes and Lebesgue integrals). To cover these topics inthe manner we cover other material would render the book more like a reference book than a text thatcould reasonably be covered in a year. Students who have completed this book will be in a good positionto study such topics at rigorous levels.We include, however, a chapter on metric spaces. We do this for two reasons: to offer a more generalframework for viewing concepts treated in earlier chapters, and to illustrate how the abstract viewpoint canbe applied to solving concrete problems. The metric space presentation in Chapter 13 can be consideredmore advanced as the reader would require a reasonable level of preparation. Even so, it is more readable andaccessible than many other presentations of metric space theory, as we have prepared it with the assumptionthat the student has just the minimal background. For example, it is easier than the corresponding chapterin our graduate level text (Real Analysis, Prentice Hall, 1997) in which the student is ex

3.12 Challenging Problems for Chapter 3 206 Notes 211 4 SETS OF REAL NUMBERS 217 4.1 Introduction 217 4.2 Points 218 4.2.1 Interior Points 219 4.2.2 Isolated Points 221 4.2.3 Points of Accumulation 222 4.2.4 Boundary Points 223 4.3 Sets 226 4.3.1 Closed Sets 227 4.3.2 Open Sets 228 4.4 Elementary To

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