Table Of ContentProblem Books in Mathematics
Antonio Caminha Muniz Neto
An Excursion
through Elementary
Mathematics,
Volume I
Real Numbers and Functions
Problem Books in Mathematics
SeriesEditor:
PeterWinkler
DepartmentofMathematics
DartmouthCollege
Hanover,NH03755
USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/714
Antonio Caminha Muniz Neto
An Excursion through
Elementary Mathematics,
Volume I
Real Numbers and Functions
123
AntonioCaminhaMunizNeto
Mathematics
UniversidadeFederaldoCeará
Fortaleza,Ceará,Brazil
ISSN0941-3502 ISSN2197-8506 (electronic)
ProblemBooksinMathematics
ISBN978-3-319-53870-9 ISBN978-3-319-53871-6 (eBook)
DOI10.1007/978-3-319-53871-6
LibraryofCongressControlNumber:2017933290
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To Gabrieland Isabela,mymostbeautiful
theorems.
To myteacher ValdenísioBezerra,in
memorian
Preface
This is the first of a series of three volumes (the other ones being [4] and [5])
devoted to the mathematics of mathematical olympiads. Generally speaking, they
are somewhat expanded versions of a collection of six volumes, first published
in Portuguese by the Brazilian Mathematical Society in 2012 and currently in its
secondedition.
The material collected here and in the other two volumes is based on course
notesthatevolvedovertheyearssince1991,whenI firstbegancoachingstudents
of Fortaleza to the Brazilian Mathematical Olympiad and to the International
MathematicalOlympiad.Some10yearsago,preliminaryversionsofthePortuguese
texts also served as textbooksfor several editions of summer courses delivered at
UFCtomathteachersoftheCapeVerdeRepublic.
All volumes were carefully planned to be a balanced mixture of a smooth and
self-containedintroductiontothe fascinatingworldofmathematicalcompetitions,
aswellastoserveastextbooksforstudentsandinstructorsinvolvedwithmathclubs
forgiftedhighschoolstudents.
Upon writing the books, I have stuck myself to an invaluable advice of the
eminent Hungarian-Americanmathematician George Pólya, who used to say that
one cannot learn mathematics without getting one’s hands dirty. That’s why, in
several points throughout the text, I left to the reader the task of checking minor
aspectsofmoregeneraldevelopments.Theseappeareitherassmallomitteddetails
in proofs or as subsidiary extensions of the theory. In this last case, I sometimes
refer the reader to specific problemsalong the book, which are marked with an *
andwhosesolutionsareconsideredtobeanessentialpartofthetext.Ingeneral,in
eachsection,Icollectalistofproblems,carefullychoseninthedirectionofapplying
thematerialandideaspresentedinthetext.Dozensofthemaretakenfromformer
editionsofmathematicalcompetitionsandrangefromthealmostimmediatetoreal
challengingones.Regardlessoftheirlevelofdifficulty,weprovidegeneroushints,
orevencompletesolutions,tovirtuallyallofthem.
This first volume concentrates on real numbers, elementary algebra, and real
functions.Thebookstartswithanon-axiomaticdiscussionofthemostelementary
propertiesofrealnumbers,followedbyadetailedstudyofbasicalgebraicidentities,
vii
viii Preface
equationsandsystemsofequations,elementarysequences,mathematicalinduction,
andthebinomialtheorem.Thesepavethewayforaninitialpresentationofalgebraic
inequalitieslikethatbetweenthearithmeticandgeometricmeans,aswellasthose
of Cauchy, Chebyshev, and Abel. We then run through an exhaustive elementary
studyoffunctionsthatculminateswith a first lookatimplicitlydefinedfunctions.
This is followed by a second look on real numbers, focusing on the concept of
convergence for sequences and series of reals. We then return to functions, this
time to successively develop, in detail, the basics of continuity, differentiability,
and integrability. Along the way, the text stays somewhere between a thorough
calculuscourseandanintroductoryanalysisone.Lotsofinterestingexamplesand
importantapplicationsarepresentedthroughout.Wheneverpossible(ordesirable),
the examplesare taken from mathematicalcompetitions,whereas the applications
vary from the proof and several applications of Jensen’s convexity inequality to
Lambert’stheoremontheirrationalityof(cid:2)andStirling’sformulaontheasymptotic
behaviorof nŠ. Thetextendswith a chapteron sequencesand seriesof functions,
where,amongotherinterestingtopics,weconstructanexampleofacontinuousand
nowhere differentiable function,develop the rudiments of the generating function
method,anddiscussWeierstrass’ approximationtheoremandtherudimentsofthe
theoryofFourierseries.
Several people and institutions contributed throughout the years for my effort
of turning a bunch of handwritten notes into these books. The State of Ceará
Mathematical Olympiad, created by the Mathematics Department of the Federal
University of Ceará (UFC) back in 1980 and now in its 36th edition, has since
then motivated hundreds of youngsters of Fortaleza to deepen their studies of
mathematics.Iwasonesuchstudentinthelate1980s,andmyinvolvementwiththis
competitionandwiththeBrazilianMathematicalOlympiadafewyearslaterhada
decisive influenceon my choice of career. Throughoutthe 1990s,I had the honor
of coaching several brilliant students of Fortaleza to the Brazilian Mathematical
Olympiad.SomeofthementeredBrazilianteamstotheIMOorotherinternational
competitions, and their doubts, comments, and criticisms were of great help in
shapingmyviewonmathematicalcompetitions.Inthissense,sincerethanksgoto
JoãoLuizFalcão,RoneyCastro,MarceloOliveira,MarcondesFrançaJr.,Marcelo
C. de Souza, Eduardo Balreira, Breno Falcão, Fabrício Benevides, Rui Vigelis,
DanielSobreira,SamuelFeitosa,DaviMáximoNogueira,andYuriLima.
Professor João Lucas Barbosa, upon inviting me to write the textbooks to the
Amílcar Cabral Educational Cooperation Project with Cape Verde Republic, had
unconsciouslyprovidedmewiththemotivationtocompletethePortugueseversion
of these books. The continuoussupportof Professor Hilário Alencar, presidentof
theBrazilianMathematicalSocietywhenthePortugueseeditionwasfirstpublished,
was also of great importance for me. Special thanks go to professors Abdênago
Barros and Fernanda Camargo, my colleagues at the Mathematics Department of
UFC, who had made quite useful comments on the Portuguese editions, which
were incorporated in the text in a way or another; they had also read the entire
Englishversionandhelpedmeinimprovingitinanumberofways.Ifitweren’tfor
myeditoratSpringer-Verlag,Mr.RobinsondosSantos,Ialmostsurelywouldnot
Preface ix
havehadthecouragetoembracethetaskoftranslatingmorethan1500pagesfrom
PortugueseintoEnglish.IacknowledgeallthestaffofSpringerinvolvedwiththis
projectinhisname.
Finally,andmostly,Iwouldlike toexpressmydeepestgratitudetomyparents
Antonioand Rosemary,my wife Monica, and our kidsGabriel and Isabela. From
early childhood,my parentshave always called my attention to the importanceof
a solid education, having done their best for me and my brothers to attend the
best possible schools. My wife and kids filled our home with the harmony and
softness I needed to get to endure on several months of solitary nights of work
whiletranslatingthisbook.
Fortaleza,Brazil AntonioCaminhaMunizNeto
December2016
Contents
1 TheSetofRealNumbers .................................................. 1
1.1 ArithmeticinR ..................................................... 3
1.2 TheOrderRelationinR............................................ 7
1.3 CompletenessoftheRealNumberSystem........................ 12
1.4 TheGeometricRepresentation..................................... 15
2 AlgebraicIdentities,EquationsandSystems............................ 19
2.1 AlgebraicIdentities................................................. 19
2.2 TheModulusofaRealNumber.................................... 27
2.3 AFirstLookatPolynomialEquations............................. 33
2.4 LinearSystemsandElimination ................................... 45
2.5 Miscellaneous....................................................... 54
3 ElementarySequences ..................................................... 61
3.1 Progressions......................................................... 61
3.2 LinearRecurrencesofOrders2and3 ............................. 70
3.3 The†and…Notations ............................................ 78
4 InductionandtheBinomialFormula .................................... 87
4.1 ThePrincipleofMathematicalInduction.......................... 87
4.2 BinomialNumbers.................................................. 98
4.3 TheBinomialFormula.............................................. 104
5 ElementaryInequalities.................................................... 111
5.1 TheAM-GMInequality............................................ 111
5.2 Cauchy’sInequality................................................. 123
5.3 MoreonInequalities................................................ 128
6 TheConceptofFunction .................................................. 143
6.1 DefinitionsandExamples .......................................... 143
6.2 Monotonicity,ExtremaandImage................................. 153
6.3 CompositionofFunctions.......................................... 163
6.4 InversionofFunctions.............................................. 172
xi
xii Contents
6.5 DefiningFunctionsImplicitly...................................... 176
6.6 GraphsofFunctions ................................................ 185
6.7 TrigonometricFunctions ........................................... 195
7 MoreonRealNumbers .................................................... 201
7.1 SupremumandInfimum............................................ 201
7.2 LimitsofSequences ................................................ 208
7.3 Kronecker’sLemma ................................................ 221
7.4 SeriesofRealNumbers............................................. 228
8 ContinuousFunctions...................................................... 245
8.1 TheConceptofContinuity......................................... 245
8.2 SequentialContinuity............................................... 256
8.3 TheIntermediateValueTheorem .................................. 264
9 LimitsandDerivatives..................................................... 275
9.1 SomeHeuristicsI................................................... 275
9.2 LimitsofFunctions................................................. 277
9.3 BasicPropertiesofDerivatives..................................... 292
9.4 ComputingDerivatives ............................................. 302
9.5 Rôlle’sTheoremandApplications................................. 312
9.6 TheFirstVariationofaFunction................................... 318
9.7 TheSecondVariationofaFunction................................ 328
9.8 SketchingGraphs................................................... 339
10 Riemann’sIntegral......................................................... 347
10.1 SomeHeuristicsII .................................................. 347
10.2 TheConceptofIntegral ............................................ 351
10.3 Riemann’sTheoremandSomeRemarks.......................... 360
10.4 OperatingwithIntegrableFunctions............................... 368
10.5 TheFundamentalTheoremofCalculus ........................... 381
10.6 TheChangeofVariablesFormula ................................. 392
10.7 LogarithmsandExponentials ...................................... 398
10.8 Miscellaneous....................................................... 414
10.9 ImproperIntegration................................................ 426
10.10 TwoImportantApplications........................................ 437
11 SeriesofFunctions.......................................................... 447
11.1 TaylorSeries ........................................................ 447
11.2 SeriesofFunctions.................................................. 456
11.3 PowerSeries ........................................................ 472
11.4 SomeApplications.................................................. 486
11.5 AGlimpseonAnalyticFunctions.................................. 499
