Authored by Hemen Dutta, Pinnangudi N. Natarajan, and Yeol Je Cho Concise Introduction to Basic Real Analysis Authored by Hemen Dutta, Pinnangudi N. Natarajan, and Yeol Je Cho Concise Introduction to Basic Real Analysis CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 (cid:13)c 2020byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-1-138-61246-4(Hardback) Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources.Rea- sonableeffortshavebeenmadetopublishreliabledataandinformation,buttheauthorand publishercannotassumeresponsibilityforthevalidityofallmaterialsortheconsequences of their use. 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CCC is a not-for-profitorganizationthatprovideslicensesandregistrationforavarietyofusers.For organizations that have been granted a photocopy license by the CCC, a separate system ofpaymenthasbeenarranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks,andareusedonlyforidentificationandexplanationwithoutintenttoinfringe. Library of Congress Cataloging-in-Publication Data Names:Dutta,Hemen,1981-author.|Natarajan,PinnangudiN.,author.| Cho,YeolJe,author Title:Conciseintroductiontobasicrealanalysis/byH.Dutta,P.N. Natarajan,andY.J.Cho. Description:BocaRaton:CRCPress,Taylor&FrancisGroup,2019. Identifiers:LCCN2019007146|ISBN9781138612464(hardback:alk.paper)| ISBN9780429464676(e-book) Subjects:LCSH:Functionsofrealvariables–Textbooks.|Mathematical analysis–Textbooks.|Numbers,Real–Textbooks. Classification:LCCQA331.5.D882019|DDC515/.88–dc23 LCrecordavailableathttps://lccn.loc.gov/2019007146 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface ix Authors xi 1 Review of Set Theory 1 1.1 Introduction and Notations . . . . . . . . . . . . . . . . . . . 1 1.2 Ordered Pairs and Cartesian Product . . . . . . . . . . . . . 2 1.3 Relations and Functions . . . . . . . . . . . . . . . . . . . . . 2 1.4 Countable and Uncountable Sets . . . . . . . . . . . . . . . . 4 1.5 Set Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 The Real Number System 13 2.1 Field Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Order Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Geometrical Representation of Real Numbers and Intervals . 15 2.4 Integers, Rational Numbers, and Irrational Numbers . . . . . 16 2.5 Upper Bounds, Least Upper Bound or Supremum, the Completeness Axiom, Archimedean Property of R . . . . . . 16 2.6 Infinite Decimal Representation of Real Numbers . . . . . . 18 2.7 Absolute Value, Triangle Inequality, Cauchy–Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.8 Extended Real Number System R∗ . . . . . . . . . . . . . . . 23 2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Sequences and Series of Real Numbers 25 3.1 Convergent and Divergent Sequences of Real Numbers . . . . 25 3.2 Limit Superior and Limit Inferior of a Sequence of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Infinite Series of Real Numbers . . . . . . . . . . . . . . . . . 28 3.4 Convergence Tests for Infinite Series . . . . . . . . . . . . . . 34 3.5 Rearrangements of Series . . . . . . . . . . . . . . . . . . . . 37 3.6 Riemann’sTheoremonConditionallyConvergentSeriesofReal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.7 Cauchy Multiplications of Series . . . . . . . . . . . . . . . . 39 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 v vi Contents 4 Metric Spaces – Basic Concepts, Complete Metric Spaces 45 4.1 Metric and Metric Spaces . . . . . . . . . . . . . . . . . . . . 45 4.2 Point Set Topology in Metric Spaces . . . . . . . . . . . . . . 46 4.3 Convergent and Divergent Sequences in a Metric Space . . . 53 4.4 Cauchy Sequences and Complete Metric Spaces . . . . . . . 54 4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Limits and Continuity 61 5.1 The Limit of Functions . . . . . . . . . . . . . . . . . . . . . 61 5.2 Algebras of Limits . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Right-Hand and Left-Hand Limits . . . . . . . . . . . . . . . 66 5.4 Infinite Limits and Limits at Infinity . . . . . . . . . . . . . 69 5.5 Certain Important Limits . . . . . . . . . . . . . . . . . . . . 70 5.6 Sequential Definition of Limit of a Function . . . . . . . . . . 71 5.7 Cauchy’s Criterion for Finite Limits . . . . . . . . . . . . . . 72 5.8 Monotonic Functions . . . . . . . . . . . . . . . . . . . . . . 73 5.9 The Four Functional Limits at a Point . . . . . . . . . . . . 75 5.10 Continuous and Discontinuous Functions . . . . . . . . . . . 75 5.11 Some Theorems on the Continuity . . . . . . . . . . . . . . . 80 5.12 Properties of Continuous Functions . . . . . . . . . . . . . . 83 5.13 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . 85 5.14 Continuity and Uniform Continuity in Metric Spaces . . . . 88 5.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 6 Connectedness and Compactness 99 6.1 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.2 The Intermediate Value Theorem . . . . . . . . . . . . . . . 105 6.3 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.4 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.5 The Finite Intersection Property . . . . . . . . . . . . . . . . 114 6.6 The Heine–Borel Theorem . . . . . . . . . . . . . . . . . . . 116 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7 Differentiation 123 7.1 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 123 7.2 The Differential Calculus . . . . . . . . . . . . . . . . . . . . 126 7.3 Properties of Differentiable Functions . . . . . . . . . . . . . 132 7.4 The L’Hospital Rule . . . . . . . . . . . . . . . . . . . . . . . 138 7.5 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 147 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8 Integration 157 8.1 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . 157 8.2 Properties of the Riemann Integral . . . . . . . . . . . . . . 168 8.3 The Fundamental Theorems of Calculus . . . . . . . . . . . . 174 8.4 The Substitution Theorem and Integration by Parts . . . . . 179 Contents vii 8.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . 181 8.6 The Riemann–Stieltjes Integral . . . . . . . . . . . . . . . . . 187 8.7 Functions of Bounded Variation . . . . . . . . . . . . . . . . 196 8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 9 Sequences and Series of Functions 213 9.1 The Pointwise Convergence of Sequences of Functions and the Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . 213 9.2 The Uniform Convergence and the Continuity, the Cauchy Criterion for the Uniform Convergence . . . . . . . . . . . . 215 9.3 The Uniform Convergence of Infinite Series of Functions . . 217 9.4 The Uniform Convergence of Integrations and Differentiations 219 9.5 TheEquicontinuousFamilyofFunctionsandtheArzela-Ascoli Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 9.6 Dirichlet’s Test for the Uniform Convergence . . . . . . . . . 224 9.7 The Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . 225 9.8 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . 227 9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 Bibliography 235 Index 237 Preface ThecontentsandapproachtothesubjectRealAnalysisadoptedinthisbook is expected to be useful for undergraduate and graduate students in under- standingthebasicconcepts.Thebookattemptstoprovideareasonableintro- duction to basic topics in real analysis and to make the subject digestible to learners. Readers will find solved examples and chapter-end exercise in each chapter including hints for solution. The book contains nine chapters and is organized as follows. In the chapter “Review of Set Theory”, we introduce basic concepts ofsettheory.WedefineCartesianproduct,relationsandfunctions,countable anduncountablesets.Weprovethatthesetofallrealnumbersisuncountable and the set of all rational numbers is countable. We then introduce the set algebra, thereby defining operations on sets such as union, intersection, and complement, and prove their properties. Inthechapter“TheRealNumberSystem”,wediscusstherealnumber system.Wetakerealnumbersforgrantedsatisfyingcertainaxiomsfromwhich furtherpropertiesarederived.Inthisdirection,weintroducethefieldaxioms, the order axioms, and the completeness axiom. We then proceed to derive several useful properties of real numbers. The chapter “Sequences and Series of Real Numbers” is devoted to a systematic study of sequences and infinite series of real numbers. In the contextofrealnumbers,weintroducetheconceptsofCauchysequences,limit superior and limit inferior, convergent and divergent sequences, convergent anddivergentseries,absolutelyconvergentandconditionallyconvergentseries and study their properties. We then prove some convergence tests for infinite series.WefurtherdiscussrearrangementofinfiniteseriesandproveRiemann’s theorem on conditionally convergent series. Finally, we discuss the Cauchy multiplication of infinite series. In the chapter “Metric Spaces – Basic Concepts, Complete Metric Spaces”,weintroducemetricspacesandrelatedbasicconcepts.Wethendis- cuss complete metric spaces. We prove the Bolzano–Weierstrass theorem and the Cantor intersection theorem. We then introduce Cauchy sequences, con- vergentanddivergentsequencesinametricspaceanddiscusstheirproperties in detail. In the chapter “Limits and Continuities”, we first discuss limit of a function, right-hand limit, left-hand limit, infinite limits, limits at infinity, sequential definition of limit, Cauchy’s criterion for finite limits and present several results covering various properties of limits with examples. Next we ix