The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs

201 Pages·2017·1.64 MB·English

by  Grinberg

The Real Analysis Lifesaver

Raffi Grinberg
PRINCETON UNIVERSITY PRESS
Princeton and Oxford
Copyright(cid:2)c 2017byPrincetonUniversityPress
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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