The Real Analysis Lifesaver Raffi Grinberg PRINCETON UNIVERSITY PRESS Princeton and Oxford Copyright(cid:2)c 2017byPrincetonUniversityPress PublishedbyPrincetonUniversityPress,41WilliamStreet, Princeton,NewJersey08540 IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet, Woodstock,OxfordshireOX201TR press.princeton.edu JacketillustrationbyDimitriKaretnikov JacketgraphicscourtesyofShutterstock AllRightsReserved LibraryofCongressCataloging-in-PublicationData Names:Grinberg,Raffi,1990– Title:Therealanalysislifesaver:allthetoolsyouneedtounderstand proofs/RaffiGrinberg. Description:Princeton:PrincetonUniversityPress,2016.|Series:A Princetonlifesaverstudyguide|Includesbibliographicalreferencesandindex. Identifiers:LCCN2016020708|ISBN9780691173870(hardcover:alk.paper)| ISBN9780691172934(pbk.:alk.paper) Subjects:LCSH:Mathematicalanalysis.|Functionsofrealvariables.| Numbers,Real. Classification:LCCQA299.8.G752016|DDC515/.8—dc23LCrecord availableathttps://lccn.loc.gov/2016020708 BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinTimesNewRomanwithStencilandAvantGarde Printedonacid-freepaper.∞ TypesetbyNovaTechsetPvtLtd,Bangalore,India PrintedintheUnitedStatesofAmerica 13579108642 C o n t e n t s Although I recommend that you read all of the chapters in order to cover a typical realanalysiscurriculum(andbecausethematerialbuildsonitself),itispossibleto takea“fastestpath”throughthisbookusingonlythechaptersmarkedwitha*. Preliminaries 1 1 Introduction 3 2 BasicMathandLogic* 6 3 SetTheory* 14 RealNumbers 25 4 LeastUpperBounds* 27 5 TheRealField* 35 6 ComplexNumbersandEuclideanSpaces 46 Topology 59 7 Bijections 61 8 Countability 68 9 TopologicalDefinitions* 79 10 ClosedandOpenSets* 90 11 CompactSets* 98 12 TheHeine-BorelTheorem* 108 13 PerfectandConnectedSets 117 Sequences 127 14 Convergence* 129 15 LimitsandSubsequences* 138 16 CauchyandMonotonicSequences* 148 17 SubsequentialLimits 157 18 SpecialSequences 166 19 Series* 174 20 Conclusion 183 Acknowledgments 187 Bibliography 189 Index 191 Preliminaries C H A P T E R 1 Introduction Slowdownthere,hotshot.Iknowyou’resmart—youmighthavealwaysbeengoodwith numbers,youmighthaveacedcalculus—butIwantyoutoslowdown.Realanalysisis an entirely different animal from calculus or even linear algebra. Besides the fact that it’sjustplainharder,thewayyoulearnrealanalysisisnotbymemorizingformulasor algorithmsandpluggingthingsin.Rather,youneedtoreadandrereaddefinitionsand proofsuntilyouunderstandthelargerconceptsatwork,soyoucanapplythoseconcepts inyourownproofs.Thebestwaytogetgoodatthisistotakeyourtime;readslowly, writeslowly,andthinkcarefully. WhatfollowsisashortintroductionaboutwhyIwrotethisbookandhowyoushould goaboutreadingit. Why I Wrote This Book Realanalysisishard.Thistopicisprobablyyourintroductiontoproof-basedmathemat- ics,whichmakesitevenharder.ButIverymuchbelievethatanyonecanlearnanything, aslongasitisexplainedclearlyenough. I struggled with my first real analysis course. I constantly felt like I was my own teacher and wished there was someone who could explain things to me in a clear, linear fashion. The fact that I struggled—and eventually pulled through—makes me anexcellentcandidatetobeyourguide.Ieasilyrecallwhatitwasliketoseethisstuff forthefirsttime.Irememberwhatconfusedme,whatwasneverreallyclear,andwhat stumped me. Inthis book, I hope I can preempt mostof your questions by giving you theexplanationsIwouldhavemostlikedtohaveseen. My course used the textbook Principles of Mathematical Analysis, 3rd edition, by Walter Rudin (also known as Baby Rudin, or That Grueling Little Blue Book). It is usuallyconsideredtheclassic,standardrealanalysistext.IappreciateRudinnow—his book is well organized and concise. But I can tell you that when I used it to learn the materialforthefirsttime,itwasaslog.Itneverexplainsanything!Rudinlistsdefinitions without giving examples and writes polished proofs without telling you how he came upwiththem. Don’tgetmewrong:havingtofigurethingsoutforyourselfcanbeoftremendous value. Being challenged to understand why things work—without linear steps handed toyouonasilverplatter—makesyouabetterthinkerandabetterlearner.ButIbelieve