Understanding Real Analysis Second Edition Paul Zorn St. Olaf College Northfield, Minnesota TEXTBOOKS in MATHEMATICS Series Editors: Al Boggess and Ken Rosen PUBLISHED TITLES ABSTRACT ALGEBRA: A GENTLE INTRODUCTION Gary L. Mullen and James A. Sellers ABSTRACT ALGEBRA: AN INTERACTIVE APPROACH, SECOND EDITION William Paulsen ABSTRACT ALGEBRA: AN INQUIRY-BASED APPROACH Jonathan K. Hodge, Steven Schlicker, and Ted Sundstrom ADVANCED LINEAR ALGEBRA Hugo Woerdeman ADVANCED LINEAR ALGEBRA Nicholas Loehr ADVANCED LINEAR ALGEBRA, SECOND EDITION Bruce Cooperstein APPLIED ABSTRACT ALGEBRA WITH MAPLE™ AND MATLAB®, THIRD EDITION Richard Klima, Neil Sigmon, and Ernest Stitzinger APPLIED DIFFERENTIAL EQUATIONS: THE PRIMARY COURSE Vladimir Dobrushkin APPLIED DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS Vladimir Dobrushkin APPLIED FUNCTIONAL ANALYSIS, THIRD EDITION J. Tinsley Oden and Leszek Demkowicz A BRIDGE TO HIGHER MATHEMATICS Valentin Deaconu and Donald C. Pfaff COMPUTATIONAL MATHEMATICS: MODELS, METHODS, AND ANALYSIS WITH MATLAB® AND MPI, SECOND EDITION Robert E. White A CONCRETE INTRODUCTION TO REAL ANALYSIS, SECOND EDITION Robert Carlson A COURSE IN DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE PROBLEMS, SECOND EDITION Stephen A. Wirkus, Randall J. Swift, and Ryan Szypowski A COURSE IN ORDINARY DIFFERENTIAL EQUATIONS, SECOND EDITION Stephen A. Wirkus and Randall J. Swift PUBLISHED TITLES CONTINUED DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE, SECOND EDITION Steven G. Krantz DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE WITH BOUNDARY VALUE PROBLEMS Steven G. Krantz DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES, THIRD EDITION George F. Simmons DIFFERENTIAL EQUATIONS WITH MATLAB®: EXPLORATION, APPLICATIONS, AND THEORY Mark A. McKibben and Micah D. Webster DISCOVERING GROUP THEORY: A TRANSITION TO ADVANCED MATHEMATICS Tony Barnard and Hugh Neill DISCRETE MATHEMATICS, SECOND EDITION Kevin Ferland ELEMENTARY DIFFERENTIAL EQUATIONS Kenneth Kuttler ELEMENTARY NUMBER THEORY James S. Kraft and Lawrence C. Washington THE ELEMENTS OF ADVANCED MATHEMATICS: FOURTH EDITION Steven G. Krantz ESSENTIALS OF MATHEMATICAL THINKING Steven G. Krantz EXPLORING CALCULUS: LABS AND PROJECTS WITH MATHEMATICA® Crista Arangala and Karen A. Yokley EXPLORING GEOMETRY, SECOND EDITION Michael Hvidsten EXPLORING LINEAR ALGEBRA: LABS AND PROJECTS WITH MATHEMATICA® Crista Arangala EXPLORING THE INFINITE: AN INTRODUCTION TO PROOF AND ANALYSIS Jennifer Brooks GRAPHS & DIGRAPHS, SIXTH EDITION Gary Chartrand, Linda Lesniak, and Ping Zhang INTRODUCTION TO ABSTRACT ALGEBRA, SECOND EDITION Jonathan D. H. Smith INTRODUCTION TO ANALYSIS Corey M. Dunn PUBLISHED TITLES CONTINUED INTRODUCTION TO MATHEMATICAL PROOFS: A TRANSITION TO ADVANCED MATHEMATICS, SECOND EDITION Charles E. Roberts, Jr. INTRODUCTION TO NUMBER THEORY, SECOND EDITION Marty Erickson, Anthony Vazzana, and David Garth INVITATION TO LINEAR ALGEBRA David C. Mello LINEAR ALGEBRA, GEOMETRY AND TRANSFORMATION Bruce Solomon MATHEMATICAL MODELING FOR BUSINESS ANALYTICS William P. Fox MATHEMATICAL MODELLING WITH CASE STUDIES: USING MAPLE™ AND MATLAB®, THIRD EDITION B. Barnes and G. R. Fulford MATHEMATICS IN GAMES, SPORTS, AND GAMBLING–THE GAMES PEOPLE PLAY, SECOND EDITION Ronald J. Gould THE MATHEMATICS OF GAMES: AN INTRODUCTION TO PROBABILITY David G. Taylor A MATLAB® COMPANION TO COMPLEX VARIABLES A. David Wunsch MEASURE AND INTEGRAL: AN INTRODUCTION TO REAL ANALYSIS, SECOND EDITION Richard L. Wheeden MEASURE THEORY AND FINE PROPERTIES OF FUNCTIONS, REVISED EDITION Lawrence C. Evans and Ronald F. Gariepy NUMERICAL ANALYSIS FOR ENGINEERS: METHODS AND APPLICATIONS, SECOND EDITION Bilal Ayyub and Richard H. McCuen ORDINARY DIFFERENTIAL EQUATIONS: AN INTRODUCTION TO THE FUNDAMENTALS Kenneth B. Howell PRINCIPLES OF FOURIER ANALYSIS, SECOND EDITION Kenneth B. Howell REAL ANALYSIS AND FOUNDATIONS, FOURTH EDITION Steven G. Krantz RISK ANALYSIS IN ENGINEERING AND ECONOMICS, SECOND EDITION Bilal M. Ayyub SPORTS MATH: AN INTRODUCTORY COURSE IN THE MATHEMATICS OF SPORTS SCIENCE AND PUBLISHED TITLES CONTINUED SPORTS ANALYTICS Roland B. Minton A TOUR THROUGH GRAPH THEORY Karin R. Saoub TRANSITION TO ANALYSIS WITH PROOF Steven G. Krantz TRANSFORMATIONAL PLANE GEOMETRY Ronald N. Umble and Zhigang Han CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20171025 International Standard Book Number-13: 978-1-1380-3301-6 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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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 1 Preliminaries: Numbers,Sets,Proofs,andBounds 1 1.1 Numbers101:TheVeryBasics . . . . . . . . . . . . . . . . . 1 1.2 Sets101:GettingStarted . . . . . . . . . . . . . . . . . . . . 10 1.3 Sets102:TheIdeaofaFunction . . . . . . . . . . . . . . . . 19 1.4 Proofs101:ProofsandProof-Writing . . . . . . . . . . . . . 32 1.5 TypesofProof. . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.6 Sets103:FiniteandInfiniteSets;Cardinality . . . . . . . . . 52 1.7 Numbers102:AbsoluteValues . . . . . . . . . . . . . . . . . 62 1.8 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1.9 Numbers103:Completeness . . . . . . . . . . . . . . . . . . 74 2 SequencesandSeries 83 2.1 SequencesandConvergence . . . . . . . . . . . . . . . . . . 83 2.2 WorkingwithSequences . . . . . . . . . . . . . . . . . . . . 96 2.3 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.4 CauchySequences . . . . . . . . . . . . . . . . . . . . . . . 114 2.5 Series101:BasicIdeas . . . . . . . . . . . . . . . . . . . . . 120 2.6 Series102:TestingforConvergenceandEstimatingLimits . . 134 2.7 Limsupandliminf:AGuidedDiscovery . . . . . . . . . . . 146 3 LimitsandContinuity 149 3.1 LimitsofFunctions . . . . . . . . . . . . . . . . . . . . . . . 149 3.2 ContinuousFunctions . . . . . . . . . . . . . . . . . . . . . . 162 3.3 WhyContinuityMatters: ValueTheorems . . . . . . . . . . . 172 3.4 UniformContinuity . . . . . . . . . . . . . . . . . . . . . . . 182 3.5 TopologyoftheRealNumbers . . . . . . . . . . . . . . . . . 190 3.6 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . 199 4 Derivatives 207 4.1 DefiningtheDerivative . . . . . . . . . . . . . . . . . . . . . 207 4.2 CalculatingDerivatives . . . . . . . . . . . . . . . . . . . . . 219 4.3 TheMeanValueTheorem . . . . . . . . . . . . . . . . . . . 230 4.4 SequencesandSeriesofFunctions . . . . . . . . . . . . . . . 239 vii viii CONTENTS 4.5 TaylorSeriesandTaylor’sTheorem:AGuidedDiscovery. . . 252 5 Integrals 257 5.1 TheRiemannIntegral:DefinitionandExamples . . . . . . . . 257 5.2 PropertiesoftheIntegral . . . . . . . . . . . . . . . . . . . . 268 5.3 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 5.4 SomeFundamentalTheorems . . . . . . . . . . . . . . . . . 292 SelectedSolutions 299 Index 335 Preface About This Edition: What’s Old and What’s New? Thissecondeditionretainsallthemainmathematicalandpedagogicalattributes and strategies of the first edition, as outlined below. For example, this edition retains—and perhaps increases—the first edition’s strong focus on helping stu- dents acquire and use mathematical language. Many students at the level envi- sionedforthistextfindthelinguisticchallengeofparsingcomplicateddefinitions and theorems just as difficult as the mathematics itself. This edition, therefore, contains additional problems and exercises that ask students to unpack, instan- tiate, or even “perturb” the statements of definitions and theorems. (See, e.g., Problem7,page 112.) This book, like any second edition, incorporatesrepair of first edition typos (andpossiblythe introductionofsome brand-newones). The narrativehasalso been“smoothed”or(intheauthor’sview)otherwiseimprovedinplacesidentified eitherbytheauthoror(thanks!)byotherusers. Followingaresomemoresignificantnewfeaturesandcontent. New sectionson topologyand compactness. Twoentirelynewsections,Sec- tions3.5and3.6,introducebasicideasoftopologyandcompactness. Somegen- eraldefinitionsandprinciplesarediscussed, butthesettingisalmostalwaysthe real line, treated (when helpful) as a metric space. Compactness is defined in terms of open covers (which pose their own linguistic challenges!) but there is strongemphasisonclosedandboundedsetsinR,viatheHeine–Boreltheorem. Thismaterialisveryoccasionallyalludedto,butnotreallydependedon,inlater sections. Inthissensethesesectionsareessentiallyself-contained,andcouldbe usedforindependentstudyorenrichmentprojects. New material on series of functions and “Taylor stuff.” Treatment of function seriesingeneralandofTaylorseriesinparticularhasbeenbeefedup,bothinSec- tion4.4andinthenewSection4.5,onTaylorseries. Section4.5joinsSection2.7 (onupperandlowerlimits)inthemodeof“guideddiscovery”.Bothsectionsare designedforstudentstoencounterasprojectsorotherformsofenrichment.Their content,thoughmathematicallyimportant,isnotexplicitlyrequiredinthesequel. ix