Problems in Real Analysis

452 Pages·2009·4.26 MB·English

Problems in Real Analysis
Teodora-Liliana T. Ra˘dulescu
Vicen¸tiu D. Ra˘dulescu
Titu Andreescu
Problems in Real Analysis
Advanced Calculus on the Real Axis
Teodora-LilianaT.Ra˘dulescu Vicen¸tiuD.Ra˘dulescu
DepartmentofMathematics SimionStoilowMathematicsInstitute
FratiiBuzestiNationalCollege RomanianAcademy
Craiova200352 Bucharest014700
Romania Romania
[e-mail] [e-mail]
TituAndreescu
SchoolofNaturalSciencesandMathematics
UniversityofTexasatDallas
Richardson,TX75080
USA
[e-mail]
ISBN:978-0-387-77378-0 e-ISBN:978-0-387-77379-7
DOI:10.1007/978-0-387-77379-7
Springer Dordrecht Heidelberg London New York
LibraryofCongressControlNumber:2009926486
MathematicsSubjectClassification(2000):00A07,26-01,28-01,40-01
©SpringerScience+BusinessMedia,LLC2009
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To understandmathematicsmeanstobeable
todo mathematics.And whatdoes itmean
doingmathematics?In thefirstplaceit
meansto beableto solvemathematical
problems.
—GeorgePo´lya(1887–1985)
Wecomenearest tothegreatwhen we are
greatin humility.
—RabindranathTagore(1861–1941)
Foreword
Thiscarefullywrittenbookpresentsanextremelymotivatingandoriginalapproach,
bymeansofproblem-solving,tocalculusontherealline,andassuch,servesasa
perfect introduction to real analysis. To achieve their goal, the authors have care-
fully selected problems that cover an impressive range of topics, all at the core
of the subject. Some problems are genuinely difficult, but solving them will be
highly rewarding, since each problem opens a new vista in the understanding of
mathematics.Thisbookisalsoperfectforself-study,sincesolutionsareprovided.
Ilikethecarewithwhichtheauthorsinterspersetheirtextwithcarefulreviews
of the backgroundmaterialneededin each chapter,thought-provokingquotations,
andhighlyinterestingandwell-documentedhistoricalnotes.Inshort,thisbookalso
makesverypleasantreading,andIamconfidentthateachofitsreaderswillenjoy
reading it as much as I did. The charm and never-ending beauty of mathematics
pervadeallitspages.
Inaddition,thislittlegemillustratestheideathatonecannotlearnmathematics
withoutsolvingdifficultproblems.Itisaworldapartfromthe“computeraddiction”
that we are unfortunatelywitnessing among the youngergenerationsof would-be
mathematicians,whousetoomuchready-madesoftwareinsteadortheirbrains,or
whostandinaweinfrontofcomputer-generatedimages,asiftheyhadbecomethe
essenceofmathematics.Assuch,itcarriesaveryusefulmessage.
Onecannothelpcomparingthisbooktoa“greatancestor,”thefamedProblems
andTheoremsinAnalysis,byPo´lyaandSzego˝,a textthathasstronglyinfluenced
generationsofanalysts.Iamconfidentthatthisbookwillhaveasimilarimpact.
HongKong,July2008 PhilippeG.Ciarlet
vii
Preface
IfIhaveseenfurtheritisbystandingontheshouldersofgiants.
—SirIsaacNewton(1642–1727),LettertoRobertHooke,1675
Mathematical analysis is central to mathematics, whether pure or applied. This
discipline arises in various mathematical models whose dependent variables vary
continuouslyandarefunctionsofoneorseveralvariables.Realanalysisdatestothe
mid-nineteenthcentury,anditsrootsgobackto the pioneeringpapersbyCauchy,
Riemann,andWeierstrass.
In1821,Cauchyestablished new requirementsofrigorin hiscelebratedCours
d’Analyse.Thequestionsheraisedarethefollowing:
– Whatisaderivativereally?Answer:alimit.
– Whatisanintegralreally?Answer:alimit.
– Whatisaninfiniteseriesreally?Answer:alimit.
Thisleadsto
– Whatisalimit?Answer:anumber.
And,finally,thelastquestion:
– Whatisanumber?
Weierstrassandhiscollaborators(Heine,Cantor)answeredthisquestionaround
1870–1872.
Ourtreatmentin thisvolumeis stronglyrelatedto the pioneeringcontributions
indifferentialcalculusbyNewton,Leibniz,Descartes,andEulerintheseventeenth
and eighteenth centuries, with mathematical rigor in the nineteenth century pro-
(cid:109)(cid:111)(cid:116)(cid:101)(cid:100)(cid:98)(cid:121)(cid:67)(cid:97)(cid:117)(cid:99)(cid:104)(cid:121)(cid:44)(cid:87)(cid:101)(cid:105)(cid:101)(cid:114)(cid:115)(cid:116)(cid:114)(cid:97)(cid:115)(cid:115)(cid:44)(cid:97)(cid:110)(cid:100)(cid:80)(cid:101)(cid:97)(cid:110)(cid:111)(cid:46)(cid:32)(cid:84)(cid:104)(cid:105)(cid:115)(cid:32)(cid:112)(cid:114)(cid:101)(cid:115)(cid:101)(cid:110)(cid:116)(cid:97)(cid:116)(cid:105)(cid:111)(cid:110)(cid:32)(cid:102)(cid:117)(cid:114)(cid:116)(cid:104)(cid:101)(cid:114)(cid:115)(cid:32)(cid:109)(cid:111)(cid:100)(cid:101)(cid:114)(cid:110)(cid:32)(cid:100)(cid:105)(cid:114)(cid:101)(cid:99)(cid:116)(cid:105)(cid:111)(cid:110)(cid:115)
(cid:105)(cid:110)(cid:32)(cid:116)(cid:104)(cid:101)(cid:32)(cid:105)(cid:110)(cid:116)(cid:101)(cid:103)(cid:114)(cid:97)(cid:108)(cid:99)(cid:97)(cid:108)(cid:99)(cid:117)(cid:108)(cid:117)(cid:115)(cid:100)(cid:101)(cid:118)(cid:101)(cid:108)(cid:111)(cid:112)(cid:101)(cid:100)(cid:98)(cid:121)(cid:82)(cid:105)(cid:101)(cid:109)(cid:97)(cid:110)(cid:110)(cid:97)(cid:110)(cid:100)(cid:68)(cid:97)(cid:114)(cid:98)(cid:111)(cid:117)(cid:120)(cid:46)
Duetothehugeimpactofmathematicalanalysis,wehaveintendedinthisbook
tobuildabridgebetweenordinaryhigh-schoolorundergraduateexercisesandmore
difficultand abstractconceptsor problemsrelated to this field. We presentin this
volumeanunusualcollectionofcreativeproblemsinelementarymathematicalanal-
ysis.Weintendtodevelopsomebasicprinciplesandsolutiontechniquesandtooffer
a systematic illustration of how to organize the natural transition from problem-
solving activity toward exploring, investigating, and discovering new results and
properties.
ix
x Preface
The aim of this volume in elementary mathematical analysis is to introduce,
throughproblems-solving,fundamentalideasandmethodswithoutlosing sightof
thecontextinwhichtheyfirstdevelopedandtheroletheyplayinscienceandpartic-
ularlyinphysicsandotherappliedsciences.Thisvolumeaimsatrapidlydeveloping
differential and integral calculus for real-valued functions of one real variable,
givingrelevancetothediscussionofsomedifferentialequationsandmaximumprin-
ciples.
Thebookismainlygearedtowardstudentsstudyingthebasicprinciplesofmath-
ematicalanalysis.However,givenitsselectionofproblems,organization,andlevel,
it would be an ideal choice for tutorial or problem-solving seminars, particularly
thosegearedtowardthePutnamexamandotherhigh-levelmathematicalcontests.
We also address this work to motivated high-school and undergraduate students.
Thisvolumeismeantprimarilyforstudentsinmathematics,physics,engineering,
andcomputerscience,but,notwithoutauthorialambition,webelieveitcanbeused
by anyonewho wants to learn elementarymathematicalanalysis by solvingprob-
lems. The book is also a must-have for instructors wishing to enrich their teach-
ingwithsomecarefullychosenproblemsandforindividualswhoareinterestedin
solvingdifficultproblemsinmathematicalanalysisontherealaxis.Thevolumeis
intendedasa challengeto involvestudentsas active participantsin the course.To
makeourworkself-contained,allchaptersincludebasicdefinitionsandproperties.
The problems are clustered by topic into eight chapters, each of them containing
bothsectionsof proposedproblemswith completesolutionsand separatesections
includingauxiliaryproblems,theirsolutionsbeingleftto ourreaders.Throughout
thebook,studentsareencouragedtoexpresstheirownideas,solutions,generaliza-
tions,conjectures,andconclusions.
The volume contains a comprehensivecollection of challenging problems, our
goal being twofold: first, to encourage the readers to move away from routine
exercises and memorized algorithms toward creative solutions and nonstandard
problem-solvingtechniques; and second, to help our readers to develop a host of
newmathematicaltoolsandstrategiesthatwillbeusefulbeyondtheclassroomand
inanumberofapplieddisciplines.Weincluderepresentativeproblemsproposedat
variousnationalor internationalcompetitions, problemsselected from prestigious
mathematicaljournals,butalsosomeoriginalproblemspublishedinleadingpubli-
cations.Thatiswhymostoftheproblemscontainedinthisbookareneitherstandard
noreasy.Thereaderswillfindbothclassicaltopicsofmathematicalanalysisonthe
realaxisandmodernones.Additionally,historicalcommentsanddevelopmentsare
presentedthroughoutthebookinordertostimulatefurtherinquiry.
Traditionally,arigorousfirstcourseorproblembookinelementarymathematical
analysisprogressesinthefollowingorder:
Sequences
Functions=⇒Continuity=⇒Differentiability=⇒Integration
Limits
Preface xi
However,thehistoricaldevelopmentofthesesubjectsoccurredinreverseorder:
Archimedes
Newton(1665)
Cauchy(1821)⇐=Weierstrass(1872)⇐= ⇐= Kepler(1615)
Leibniz(1675)
Fermat(1638)
Thisbookbringsto lifetheconnectionsamongdifferentareasofmathematical
analysis and explains how various subject areas flow from one another. The vol-
umeillustratestherichnessofelementarymathematicalanalysisasoneofthemost
classicalfieldsinmathematics.Thetopicisrevisitedfromthehigherviewpointof
universitymathematics,presentingadeeperunderstandingoffamiliarsubjectsand
anintroductiontonewandexcitingresearchfields,suchasGinzburg–Landauequa-
tions, the maximum principle, singular differential and integral inequalities, and
nonlineardifferentialequations.
The volume is divided into four parts, ten chapters, and two appendices, as
follows:
PartI.Sequences,Series,andLimits
Chapter1.Sequences
Chapter2.Series
Chapter3.LimitsofFunctions
PartII.QualitativePropertiesofContinuousandDifferentiableFunctions
Chapter4.Continuity
Chapter5.Differentiability
PartIII.ApplicationstoConvexFunctionsandOptimization
Chapter6.ConvexFunctions
Chapter7.InequalitiesandExtremumProblems
PartIV.Antiderivatives,RiemannIntegrability,andApplications
Chapter8.Antiderivatives
Chapter9.RiemannIntegrability
Chapter10.ApplicationsoftheIntegralCalculus
AppendixA.BasicElementsofSetTheory
AppendixB.TopologyoftheRealLine
Eachchapterisdividedintosections.Exercises,formulas,andfiguresarenum-
beredconsecutivelyineachsection,andwe also indicateboththechapterandthe
sectionnumbers.We haveincludedatthebeginningofchaptersandsectionsquo-
tations from the literature. They are intended to give the flavor of mathematics as
asciencewithalonghistory.Thisbookalsocontainsarichglossaryandindex,as
wellasalistofabbreviationsandnotation.
xii Preface
Keyfeaturesofthisvolume:
– contains a collection of challenging problems in elementary mathematical
analysis;
– includesincisiveexplanationsofeveryimportantideaanddevelopsilluminating
applications of many theorems, along with detailed solutions, suitable cross-
references,specifichow-tohints,andsuggestions;
– isself-containedandassumesonlyabasicknowledgebutopensthepathtocom-
petitiveresearchinthefield;
– usescompetition-likeproblemsasaplatformfortrainingtypicalinventiveskills;
– develops basic valuable techniques for solving problems in mathematical ana-
lysisontherealaxis;
– 38carefullydrawnfiguressupporttheunderstandingofanalyticconcepts;
– includes interesting and valuable historical account of ideas and methods in
analysis;
– containsexcellentbibliography,glossary,andindex.
Thebookhaselementaryprerequisites,anditisdesignedtobeusedforlecture
coursesonmethodologyofmathematicalresearchordiscoveryinmathematics.This
workisafirststeptowarddevelopingconnectionsbetweenanalysisandothermath-
ematicaldisciplines,aswellasphysicsandengineering.
The background the student needs to read this book is quite modest. Anyone
with elementary knowledge in calculus is well-prepared for almost everything to
befoundhere.Takingintoaccounttherichintroductoryblurbsprovidedwitheach
chapter,noparticularprerequisitesarenecessary,evenifadoseofmathematicalso-
phisticationisneeded.Thebookdevelopsmanyresultsthatarerarelyseen,andeven
experiencedreadersarelikelytofindmaterialthatischallengingandinformative.
Ourvisionthroughoutthisvolumeiscloselyinspiredbythefollowingwordsof
George Po´lya [90] (1945) on the role of problems and discovery in mathematics:
Infallible rules of discovery leading to the solution of all possible mathematical
problemswouldbemoredesirablethanthephilosopher’sstone,vainlysoughtbyall
alchemists.Thefirstruleofdiscoveryistohavebrainsandgoodluck.Thesecond
ruleofdiscoveryistosittightandwaittillyougetabrightidea.Thoseofuswho
have little luck and less brain sometimes sit for decades.The fact seems to be, as
Poincare´ observed,itistheman,notthemethod,thatsolvestheproblem.
Despite our best intentions, errors are sure to have slipped by us. Please let us
knowofanyyoufind.
August2008 Teodora-LilianaRa˘dulescu
Vicen¸tiuRa˘dulescu
TituAndreescu