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UndergraduateAnalysis Undergraduate Analysis AWorkingTextbook AislingMcCluskey
SeniorLecturerinMathematics NationalUniversityofIreland,Galway
BrianMcMaster
HonorarySeniorLecturer Queen’sUniversityBelfast
GreatClarendonStreet,Oxford,OX26DP, UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof OxfordUniversityPressintheUKandincertainothercountries
©AislingMcCluskeyandBrianMcMaster2018
Themoralrightsoftheauthorshavebeenasserted
FirstEditionpublishedin2018
Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe addressabove
Youmustnotcirculatethisworkinanyotherform andyoumustimposethissameconditiononanyacquirer
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress 198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
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Wededicatethisbooktoallthosepractitionersofthecraftofanalysiswhose apprenticeswehavebeenintimeslongpast,andtothecolleagueswhoinmore recentyearshavesharedwithustheirinsightsandtheirenthusiasm.
Inparticular,wesalutewithgratitudeandaffection:
SamuelVerblunsky
DerekBurgess
RalphCooper
JamesMcGrotty
DavidArmitage
TonyWickstead
ArielBlanco
RayRyan
JohnMcDermott
AMcC,BMcM,October2017
Preface Mathematicalanalysisunderpinscalculus:itisthereasonwhycalculusworks,and itprovidesatoolkitforhandlingsituationsinwhichalgorithmiccalculusdoesn’t work.Sincecalculusinitsturnunderpinsvirtuallythewholeofthemathematical sciences,analyticideaslierightattheheartofscientificendeavour,sothata confidentunderstandingoftheresultsandtechniquesthattheyinformisvaluable forawiderangeofdisciplines,bothwithinmathematicsitselfandbeyondits traditionalboundaries.
Thishasachallengingconsequenceforthosewhoparticipateinthird-level mathematicseducation:largenumbersofstudents,manyofwhomdonotregard themselvesprimarilyasmathematicians,needtostudyanalysistosomeextent;and inmanycasestheirprogrammesdonotallowthemenoughtimeandexposureto growconfidentinitsideasandtechniques.Thisprogramme-timepovertyisone ofthecircumstancesthathavegivenanalysistheunfortunatereputationofbeing strikinglymoredifficultthanothercognatedisciplines.
Aspectsofthisperceptionofdifficultyincludethe lackofintroductorygradualness generallyobservedintheliterature,andthe withoutlossofgenerality factor: experiencedanalystsarecontinuallysimplifyingtheirargumentsbysummoning upabatteryofshortcuts,estimationsandreductions-to-special-casesthatare partofthediscipline’sfolklore,butwhichthereisseldomclasstimetoteachin anyformalsense:instead,studentsareexpectedtopickuptheseideasthrough experienceofworkingonexamples.Yetthestudytimeallocatedtoanalysisin earlyundergraduateprogrammesisofteninsufficientforthiskindoflearning byosmosis.Theironicconsequenceisthatbasicanalyticexercisesarenotonly substantiallyharderforthebeginnerthanfortheprofessional,butsubstantially harderthantheyneedtobe.
Thistext,throughitscarefuldesign,emphasisandpacing,setsouttodevelop understandingandconfidenceinanalysisforfirst-yearandsecond-yearundergraduatesembarkeduponmathematicsandmathematicallyrelatedprogrammes. Keenlyawareofcontemporarystudents’diversityofmotivation,background knowledgeandtimepressures,itconsistentlystrivestoblendbeneficialaspects oftheworkbook,theformalteachingtextandtheinformalandintuitivetutorial discussion.Inparticular:
1.Itdevotesamplespaceandtimefordevelopmentofinsightandconfidencein handlingthefundamentalideasthat–ifimperfectlygrasped–canmake analysisseemmoredifficultthanitactuallyis.
2.Itfocusesonlearningthroughdoing,presentingacomprehensiveintegrated rangeofexamplesandexercises,someworkedthroughinfulldetail,some supportedbysketchsolutionsandhints,someleftopentothereader’s initiative(andsomewithonlinesolutionsaccessiblethroughthepublishers).
3.Withoutundervaluingtheabsolutenecessityofsecurelogicalargument,it legitimisestheuseofinformal,heuristic,evenimpreciseinitialexplorationsof problemsaimedatdecidinghowtotacklethem.Inthisrespectitcreatesan atmospherelikethatofanapprenticeship,inwhichthetraineeanalystcan lookovertheshoulderoftheexperiencedpractitioner,lookunderthebonnet oftheproblemandwatchtheroughworkdevelop,notingtheoccasional failuresofopeninggambitsandthetricksofthetradethatcanbemobilisedin ordertocircumventthem.
Thepricethathastobepaidforsuchanapproachisthatthebookismore verbose,sometimespositivelylong-winded,andcertainlylongerthanonethat wouldconcentratesolelyonfinalisedversionsofstandardproofsandslickmodel answers.Yetitappearstousthatsuchapriceiswellworthpaying:foronething, itisourexperiencethatatextprincipallyconsistingofstreamlined,finalised demonstrationsandsolutionscreatesinthemindofmanybeginnersamisleading anddemoralisingimpressionthatthisishowtheyareexpectedtocreatesolutions atthefirstattempt;foranother,theextramaterial–farfrombeingjustdigressional –summariseswhatwefinditnecessarytosay,timeandtimeagain,tostudentswho askuseminentlyreasonablequestionssuchas:‘HowdoIstartthis?’‘Howcanwe beexpectedtothinkofthat?’‘Whyisthatsteptrue,andwhydidyouthinkoftaking it?’Anadditionalbenefitisthatthetextwillbeeasierandquickertoread,since thethoughtfulreaderwilloftenfindanswerspromptlysuppliedtothequestions thatwouldotherwisehaveimpededprogresstothenextstep.
Especiallybecauseless-specialisedlearnerswilloftenneedtodealwithonly someofthematerialcoveredhere,wehavestreamedthepresentationintobasic andmoreadvancedchaptersand,withinthese,wehaveflaggeduprelatively specialisedtopicsandsophisticatedargumentsthatcanreasonablybeomitted withoutcompromisingoverallcomprehension.Analysisismorewelcomingtothe learnerwhohasthoroughlygraspedamodestamountofmaterialthantoonewho hasanimpreciseunderstandingofalargerbodyofknowledge.
Itiscentraltoourteachingphilosophyandtoourclassroomexperiencethat studentslearnatadeeperlevelthroughdoingthantheyevercouldthrough readingalone:despiteourintentiontopresenthereasfullanaccountofbasic analyticconcepts,resultsandtechniquesasisreasonabletosetbeforelearners whohavemanyothercompetingdemandsontheirtimeandenergy,itisonlyby activestudy,engaginginabroadrangeofexercises,thattheywillgainconfidence andempowermentinacquiringuseable,performableknowledgeandtheinsight thatdirectsit.Ouraccountisthereforeintendedasaworkingtextbook:each ideaencounteredisembeddedinworkedexamplesandinexercises–some withsolutions,somewithhelpfulhintsencouragingthereadertoexploreandto internalisethatidea.
ANotetotheInstructorxiii
ANotetotheStudentReaderxv
1Preliminaries1 1.1Realnumbers1
1.2Thebasicrulesofinequalities—achecklistofthingsyouprobablyknow already2 1.3Modulus3 1.4Floor4
2Limitofasequence—anidea,adefinition,atool5 2.1Introduction5
2.2Sequences,andhowtowritethem6 2.3Approximation10
2.4Infinitedecimals11 2.5Approximatinganarea13
2.6Asmallsliceof π 16
2.7Testinglimitsbythedefinition17
2.8Combiningsequences;thealgebraoflimits24
2.9POSTSCRIPT:toinfinity29 2.10Importantnoteon‘elementaryfunctions’35
3Interlude:differentkindsofnumbers37 3.1Sets37
3.2Intervals,maxandmin,supandinf40 3.3Denseness47
4Upanddown—increasinganddecreasingsequences53
4.1Monotonicboundedsequencesmustconverge53
4.2Induction:infinitereturnsforfiniteeffort62
4.3Recursivelydefinedsequences71
4.4POSTSCRIPT:Theepsilonticsgame—the‘fifthfactorofdifficulty’75
5Samplingasequence—subsequences77 5.1Introduction77
5.2Subsequences77
5.3Bolzano-Weierstrass:theovercrowdedinterval83
6Special(orspeciallyawkward)examples87 6.1Introduction87
6.2Importantexamplesofconvergence87
7Endlesssums—afirstlookatseries103 7.1Introduction103
7.2Definitionandeasyresults104
7.3Bigseries,smallseries:comparisontests111
7.4Theroottestandtheratiotest118
8Continuousfunctions—thedomainthinksthatthegraphisunbroken125 8.1Introduction125
8.2Aninformalviewofcontinuity127
8.3Continuityatapoint133
8.4Continuityonaset134
8.5Keytheoremsoncontinuity138
8.6Continuityoftheinverse146
9Limitofafunction153
9.1Introduction153 9.2Limitofafunctionatapoint158
10Epsilonticsandfunctions169
10.1Theepsilonticviewoffunctionlimits169 10.2Theepsilonticviewofcontinuity174 10.3One-sidedlimits177
11Infinityandfunctionlimits185
11.1Limitofafunctionas x tendstoinfinityorminusinfinity185
11.2Functionstendingtoinfinityorminusinfinity192
12Differentiation—theslopeofthegraph201 12.1Introduction201 12.2Thederivative203
12.3Upanddown,maximumandminimum:fordifferentiablefunctions213 12.4Higherderivatives223
12.5Alternativeproofofthechainrule225
13TheCauchycondition—sequenceswhosetermspacktightlytogether229
13.1Cauchyequalsconvergent229
14Moreaboutseries237
14.1Absoluteconvergence237 14.2The‘robustness’ofabsolutelyconvergentseries242 14.3Powerseries252
15Uniformcontinuity—continuity’sglobalcousin259 15.1Introduction259
15.2Uniformlycontinuousfunctions263 15.3Theboundedderivativetest272
16Differentiation—meanvaluetheorems,powerseries277 16.1Introduction277
16.2Cauchyandl’Hôpital277 16.3Taylorseries284
16.4Differentiatingapowerseries287
17Riemannintegration—areaunderagraph293 17.1Introduction293
17.2Riemannintegrability—howcloselycanrectanglesapproximateareas undergraphs?295
17.3Theintegraltheoremsweoughttoexpect305
17.4Thefundamentaltheoremofcalculus313
18Theelementaryfunctionsrevisited325 18.1Introduction325
18.2Logarithmsandexponentials325 18.3Trigonometricfunctions332
19Exercises:foradditionalpractice341
Suggestionsforfurtherreading377 Index379
ANotetotheInstructor Thefirsttwelvechapterspresenttheideasofanalysistowhichvirtuallyeveryone enrolleduponadegreepathwaywithinmathematicalscienceswillrequireexposure.Thosewhosedegreeisexplicitlyinmathematicsarelikelytoneedmostofthe rest.Ofcourse,howthismaterialisdividedacrosstheyearsoracrossthesemesters willvaryfromoneinstitutiontoanother.
Mostoftheexercisessetoutwithinthetextareprovidedwithspecimen solutionseithercomplete,outlinedorhintedat,butinthefinalchapterwehave alsoincludedasuiteofovertwohundredproblemswhichareintendedtoassistyou increatingassessmentsforyourstudentgroups.Specimensolutionstotheseare availabletoyou,butnotdirectlytoyourstudents,byapplicationtothepublishers: pleaseseethewebpagewww.oup.co.uk/companion/McCluskey&McMasterfor howtoaccessthem.
Priorknowledgethatthereadershouldhavebeforeundertakingstudyofthis materialincludesafamiliaritywithelementarycalculusandbasicmanipulative algebraincludingthebinomialtheorem,agoodintuitiveunderstandingofthe realnumbersystemincludingrationalandirrationalnumbers,basicprooftechniquesincludingproofbycontradictionandbycontraposition,verybasicset (andfunction)theory,andtheuseofsimpleinequalitiesincludingmodulus. Substantialrevisionnotesonseveralofthesetopicsareprovidedwithinthetext whereappropriate.
ANotetotheStudentReader If,asastudentofthematerialthatthisbooksetsforth,youareenrolledona courseofstudyatathird-levelinstitution,yourinstructorswillguideandpaceyou throughit.Carefulconsiderationofthefeedbacktheygiveyouontheworkyou submitwillbeveryprofitabletoyouasyoudevelopcompetenceandconfidence.
Ifyouareanindependentreader,notengagedwithsuchaninstitution’sprogrammes,weintendthatyoualsowillfindthatthetextsupportsyourendeavoursthroughitsdesign:inparticular,throughtheexpansive(almostleisurely) treatmentoftheinitialideasthatreallyneedtobethoroughlygraspedbeforeyou proceed,throughtheinformalandintuitivebackgrounddiscussionsthatseekto developafeelforconceptsthatwillworkinparallelwiththeirprecisemathematical formulations,andthroughtheexplicitinclusionofroughworkparagraphsthat allowyoutolookovertheshoulderofthemoreexperiencedpractitionerofthe craftandunderthebonnetoftheproblembeingtackled.
Inbothcases,ourstrongestadvicetoyouistoworkthrougheveryexercise asyouencounterit,andeithercheckyouransweragainstaspecimenanswer whereavailable,seeifitconvincesacolleagueorfellowstudent,orsubmititfor assessmentorfeedbackasappropriate.Nobodylearnsanalysismerelybyreading it,anymorethanyoucanlearnswimmingorcyclingjustbyreadingahow-tobook, howeverwell-intentionedorknowledgablywrittenitmaybe.Noonecanteachyou analysiswithoutyourcommitment;butyoucanchoosetolearnitand,ifyoudo, thisworkingtextbookisdesignedtohelpyoutowardssuccess.
Preliminaries 1.1 Realnumbers Youcanchoosetothinkoftherealnumbersasbeingallthepossibledecimals–finiteandinfinite,recurringandnon-recurring,positiveandnegativeandzero, wholenumbersandfractionsandsurds1andnon-surdssuchas π and e,andevery possiblecombinationofsuchobjects.Equallywell,youcanchoosetothinkof themasbeing(orbeingrepresentedby)allthepointsthatlieonacontinuous unbrokenstraightline(the realline,the realaxis)thatstretchesawayendlesslyin bothdirections.Somewhereonthatlineisapointmarked0(zero)whichseparates thepositives(onitsright)fromthenegatives(onitsleft),andpacingoutfrom zeroatregularintervalsinbothdirectionsliethewholenumbers(the integers)like distancemarkersalongthatendlessroad.
Thisisnot,ofcourse,aproperdefinitionofwhatrealnumbersare.Wearetaking whatissometimescalleda naïve viewofthesystemofrealnumbers:nothaving sufficienttimetoconstructit–todigdeeplyenoughintothelogicalfoundationsof mathematicstocomeupwithaguaranteeofitsexistence–weareinsteadseeking tohighlightthecommonconsensusonhowrealnumbersbehave,combineand compare.Thisconsensuswillalreadybeenoughtoletusstartexplainingsome basicideasinanalysis(andweshallsaymoreaboutthefinerstructureofthereal numbersinChapter3).
NothinginSection1.2islikelytostrikethestudentreaderasbeingmuchmore thancommonsense,andnorshoulditatthisstageofstudy.Nevertheless,itisall tooeasytomakemistakesin comparisonsbetweennumbers – inequalities –andit isconsequentlyimportanttokeeptheseapparentlyobviousrulesinmindandto buildupagoodmeasureofconfidenceintheiruse,especiallybecausesomany argumentsinanalysisdependuponusinginequalities.Sections1.3and1.4present acoupleofusefuloperationsonrealnumbersthatarestronglyconnectedwith inequalities.
1thatis,non-rationalnumbersinvolvingroots,suchas √2, 3 √5 1 + √2 , 10 3 √2.
UndergraduateAnalysis:AWorkingTextbook,AislingMcCluskeyandBrianMcMaster2018. ©AislingMcCluskeyandBrianMcMaster2018.Published2018byOxfordUniversityPress
1.2 Thebasicrulesofinequalities—achecklist
ofthingsyouprobablyknowalready •Eachrealnumberiseitherpositiveorzeroornegative.‘Non-negative’means positiveorzero.
• x > y and y < x bothmean x y ispositive2.
• x ≥ y and y ≤ x bothmean x y isnon-negative3.
• x < y < z meansboth x < y and y < z .Likewisefor >, ≤, ≥.
•If x < y and y < z ,then x < z .Likewisefor >, ≤, ≥
•If x ≤ y and y ≤ x,then x = y.
•If x and y aredifferentrealnumbers,thenoneofthemisgreaterthantheother, andisusuallydenoted4bymax{x, y}.
•Youcanaddanumbertoaninequalitywithoutdamagingit:
x < y ⇒ x + a < y + a.
•Youcanaddtwoinequalities:
(x < y and a < b) ⇒ x + a < y + b
•Noticehowtousethesymbol‘ ⇒ ’(pronounced implies):thelastlineis shorthandfor‘if x < y and a < b then x + a < y + b’.
•Youcanmultiplyaninequality byapositivenumber withoutdamagingit: provided a > 0,wehave x < y ⇒ ax < ay.
•Ifyoumultiplyaninequality byanegativenumber,theinequalitybecomes reversed:
providedthat a < 0,wehave x < y ⇒ ax > ay.
•Youcanmultiplytwoinequalitiesprovidedthatallthenumbersinvolvedare positive:
(0 < a < b and0 < x < y) ⇒ ax < by ; (0 < a ≤ b and0 < x ≤ y) ⇒ ax ≤ by.
•Providedthatthenumbersinvolvedarepositive,youcantakereciprocals acrossaninequality,andtheinequalitybecomesreversed: x < y ⇒ 1/x > 1/y providedthat x, y arepositive.
•Providedthatthenumbersinvolvedarepositive,youcantakesquareroots5 acrossaninequality,andtheinequalityispreserved:
x < y ⇒ √x < √y providedthat x, y arepositive.Likewiseforcuberoots, fourthrootsandsoon.
2–andarepronouncedas x isgreater/larger/biggerthan y, y isless/smallerthan x 3–andarepronouncedas x isgreaterthanorequalto y, y islessthanorequalto x 4If x = y thenmax{x, y} means x (or y,whichisthesamething).
5Recallthatthesymbol √x alwaysmeansthe non-negative squarerootof x
•‘Therearelargeintegers:’thatis,foranygivenrealnumber x wecanfindan integer n sothat n > x.
1.3 Modulus 1.3.1 Definition If x isarealnumber,wedefine6its modulus (alsocalledits absolutevalue)as |x|= thegreaterof x and x.Thatis:
•If x ≥ 0then |x|= x;
•If x < 0then |x|=−x.
Sincetheeffectofmodulusisto‘throwawaytheminusfromnegativenumbers’, thefollowingshouldbeobvious:
1.3.2 Proposition Foranyrealnumbers x, y:
• x ≤|x|, x ≤|x|,
• |− x|=|x|,
• |xy|=|x||y|,
• x y = |x| |y| providedthat y = 0,
• √x2 =|x|.
1.3.3 Thetriangleinequality Foranyrealnumbers x and y,wehave |x + y|≤|x|+|y|
Proof
Since x ≤|x| and y ≤|y|,addinggivesus x + y ≤|x|+|y|. Exactlythesamereasoninggivesus x + ( y) =−(x + y) ≤|x|+|y| Now |x + y| iseither x + y or (x + y).Sowhicheveroneitis,itis ≤|x|+|y|
Note
Itiseasytoextendthisbyinduction7todealwithanyfinitelistofnumbers,thus: |x1 + x2 + x3 + ... + xn |≤|x1 |+|x2 |+|x3 |+ ... +|xn |.
1.3.4 Thereversetriangleinequality Foranyrealnumbers x and y,wehave |x|−|y| ≤|x y|.
6Morebriefly: |x|= max{x, x}
7Wediscussthistypeofargumentindetaillaterinthetext.
Proof Usethetriangleinequalityon x = (x y) + y andweget |x|≤|x y|+|y|,from which |x|−|y|≤|x y|.
Interchange x and y,andwealsoget |y|−|x|≤|y x|=|x y|.
Now |x|−|y| iseither |x|−|y| or |y|−|x|.Sowhicheveroneitis,itis ≤|x y|.
1.4 Floor 1.4.1 Definition When x isarealnumber,wedefinethe floor of x (alsocalled the integerpart of x or,informally, xroundeddowntothenearestinteger)tobethe largestintegerthatis ≤ x.Theusualnotationforthefloorof x is x ,althoughsome bookswriteitas[x].Forinstance, 5.6 = 5, π = 3, 7 = 7, −8 1 2 =−9.
Ifyouchoosetoimaginetherealnumbersasbeingsetoutalongtherealline, withtheintegers–markedherebyheavierdots–embeddedintoitatregular intervals,thenthefollowingdiagramshouldhelpyoutopicturetherelationship between x and x
+1
Case1:when x isnotaninteger x
+1
Case2:when x itselfisaninteger
Inbothcases,theessentialinequalityconnecting x and x is
≤ x < x + 1 or,equivalently
1 < x ≤ x
Limitofasequence —anidea,adefinition, atool 2.1 Introduction Mathematicalanalysishasacquiredareputation–notentirelyjustified–for seemingmoredifficultthanotherfirst-yearundergraduatestudyareas.Weshall beginourexplorationofitbyseekingtoidentifythefactorsthathavecontributed tothisimage,andwhatwecandotoexplainoraddressthem.
Firstly,thestudyofmathematicsis cumulative toagreaterdegreethanthat ofmostdisciplines.Eachnewblockofmathematicsthatastudentencountersis builtdirectlyonother,underpinning,blocks,anditispracticallyimpossibleto achieveconfidenceinthenewwithouthavingpreviouslyidentifiedandgraspedthe oldersupportingmaterial.Nomatterhowwellyoucanimplementdifferentiation algorithms,yourchanceofsuccessfullyfindingthesecondderivativeof x4 isvery limiteduntilyou’velearnedyourthree-timestable.
Secondly,mathematicsis hard.Bythatwedonotmeanthatitisintrinsically difficult:inthissense,‘hard’istheoppositeof‘soft’,nottheoppositeof‘easy’. Learningapieceofmathematicsrequiresapreciseunderstandingoftheterms thatitinvolves,oftheargumentsthatitemploysandofthequestionsthatitseeks toanswer.Abroadappreciation,asolidgeneraloverviewofthetopic,willonits ownbeutterlyinsufficientforactualapplication. Precision ofconceptandoflogical discourse,aswellasthepreviouslymentionedcumulativeness,arethehallmarks ofadisciplinethatis‘hard’inthissense.
Yetthesetwofactorsarecommontothewholeofmathematics.Whydoes analysisinparticularhavesuchadauntingpublicimage?
Itseemstousthat,thirdly,a lackofintroductorygradualness comesintoplay here.Mosttopics,inmathematicsandelsewhere,canbeadequatelyexplainedto thebeginnerbyworkinginitiallyonsimplespecialcases.Sotheusualarenafor firststepsinlinearalgebraissomethinglikethecoordinateplane,ratherthan aninfinite-dimensionalBanachspace;Frenchlanguagelessonsdonotkickoffby handingoutatableofthecompletetensesofcommonirregularverbs.Inanalysis, however,theveryfirstconceptthatabeginnerhastomakesenseofisoneofthe mostdemanding:untilyouhaveacrispunderstandingofthenotionofthelimit ofasequence(or,amatterofsimilardifficulty,ofthesupremumofasetofreal UndergraduateAnalysis:AWorkingTextbook,AislingMcCluskeyandBrianMcMaster2018. ©AislingMcCluskeyandBrianMcMaster2018.Published2018byOxfordUniversityPress
numbers)youcanneitherreadnorcarryoutanysignificantanalyticactivity.On thecreditside,thismeansthatwecanhonestlypromisethebeginnerthatthe materialgetseasieroncewearethroughmostofChapter2–aninterestingcontrast withmanytopics,bothmathematicalandotherwise–providedalwaysthatthis firstconceptisfullyandthoroughlyunderstoodbeforewegoanyfurther.
Fourthly–andthisisanotherpointthatappliestothewholeofthediscipline, butisparticularlyrelevantjusthere–mathematicsasasubjectandmathematicians asabreedareinclinedtoprefer conciseness oververbositywhentheypresentfinal versionsoftheirwork,andtofeelmoreathomewithterse,lean,point-by-point argumentsratherthanexpansive,wordy,descriptiveaccounts.Thereare,however, somekeymomentsinanalysiswhereexpansiveratherthancompressedaccounts actuallyhelpindeliveringunderstanding,andthedefinitionofsequencelimits, rightatthestartofourstudy,isoneofthem.Itisperfectlypossibletowritedown thatdefinitioninoneline:butifwedo,mostreaderswillnotseethepointofit, willnotgraspthekindofproblemthatitissetuptoaddressandwillnotbeable tomakeeffectiveuseofiteveninquitesimpleexamples.So–withapologiestoall thosewhodon’tlikereadingessays–weseenoalternativetospendingafairbitof timeandseveralhundredwordsfillinginthebackgroundand‘thinkingoutloud’ abouthowtousethisideainapplications.Weagainreiteratethattheconceptitself isnotintrinsicallydifficult;itismerelydifferentfrommathematicalnotionsthat youhavealreadymastered,andneedsaparticularformofargumentpresentation inordertogetthebestoutofit.Wealsocommittogettingbacktoconcise,unwordyargumentsassoonasandwhereverpossible.
Withallthisinmind,weshalldevotemostofChapter2toathoroughand leisurelyexplorationofthisonesingleideathatopensthepathtoanalyticargumentsinmathematics:limitsofsequences–itsintuitivemeaning,someofthe contextsinwhichitarises,howtodefineitintermssufficientlyprecisetodo seriousmathematicswithit,andhowtohandlethatrigorousdefinitioninarange ofillustrativeexamples.Pleasekeepinmindthat,oncetheopeningchapterissafely assimilated,mostoftherestofthefirst-yearanalysissyllabusiseasier.(Bytheway, thereisafifthfactorcontributingtothewidespreadperceptionofthedifficultyof introductoryanalysis,butitconcernsitslogicalstructureratherthanitsnarrowly mathematicalcontent,soweshallsetitasideuntilsomefamiliaritywiththebasic ideahasbeengained–seeSection4.4.)
2.2 Sequences,andhowtowritethem A realsequence inmathematics,sometimesmoreproperlycalledan infinitereal sequence,isanunendinglistofrealnumbersinaparticularorder:afirstone,a second,athird,andsoonwithoutend.Inothertopicswithinmathematics,itpays tolookatunendinglistsofobjectsofotherkinds–complexnumbers,functions, sets–butforthepresentweshallrestrictourfocustorealnumbers,andusethe singleword‘sequence’alwaystomean‘realsequence’(sincenoothervarieties areunderourattention).Thesortsofsymbolsthatwewritedowntoidentifya particularsequencethatwewanttoworkwithlooklikeoneofthefollowing:
(a1 , a2 , a3 , a4 , , an , )
(a1 , a2 , a3 , a4 , ··· )
(an )n∈N
(an )n≥1
(an )
–andinmanycaseswecompletethedescriptionbysettingdownaformulafor howtocalculateeachindividualnumber an inthelist(theso-called nth term).For instance,ifwewishtotalkaboutthesequenceofallperfectsquares,thatis,all thesquaresofpositiveintegersintheirnaturalorder,thenallofthefollowingare acceptablesymbols:
(1,4,9,16, ··· , n2 , ··· )
(1,4,9,16, ··· )
(n2 )n∈N
(n2 )n≥1
(n2 )
(a1 , a2 , a3 , a4 , ··· , an , ··· ) where an = n2 foreachpositiveinteger n
(a1 , a2 , a3 , a4 , ) where an = n2 ,eachpositiveinteger n
(an )n∈N inwhich an = n2 foreach n
(an )n≥1 with an = n2 foreach n
(an ), an = n2 foreachpositiveinteger n
Itmayseemalittleirritatingthatsomanydifferentstylesofsymbolareallowed, butthisismostlytoenableustotailorthenotationweusetotheparticularproblem thatweareworkingonwithoutwritingmorethanisnecessary.Forinstance,ifthe formulafor an isassimpleas an = n2 ,thenwereallyhavenoneedforaseparate symbolforthe nth term,andwemightjustaswellwriteitas n2 allthetime;onthe otherhand,ifthe nth termissomethingascomplicatedas
thenweshallcertainlynotwanttowritethatoutmoreoftenthanisneedful,andin suchcases,havingabriefsymbolsuchas an tostandinforitwillbeaconsiderable benefitandrelief.
Althoughtheideaofdenotingasequencebyalistofitsfirstfewtermsora formulaforitsgeneralterm,wrappedupinbrackets,islittlemorethancommon sense,itwillbeimportanttousethisnotationconsistentlyandcorrectly.Sowe nowflagupafew dos and don’ts concerninghowbesttoemployit:
•Wheneveryouuseanotationlike (a1 , a2 , a3 , a4 , ··· , an , ··· ) or (a1 , a2 , a3 , a4 , ),becarefulnottoleaveoutthefinalrowofdots:becausea symbolsuchas (a1 , a2 , a3 , a4 , , an ) or (a1 , a2 , a3 , a4 ) isastandardwayto writea finite listofnumbersconsistingofonly n or,indeed,onlyfouritems, andyouwillconfusethepersonreadingyourworkifyouuseitwhenyou actuallyintendaninfinitesequence.
•Alsobecautiousaboutusingsuchasymbolas (1,4,9,16, ):however obviousitmaybetoyouthatthisintendsthesequenceofperfectsquares,there are other perfectlygoodsequenceswhosefirstfourtermsare1,4,9and16. Therefore,onlyusethisstyleofnotationifitisgenuinelyclearwhatthe ‘pattern’ofthetermsis.Notethatthesymbol (12 ,22 ,32 ,42 , ··· ) makesthis patternquiteunambiguous.
•Alwaystakecarenottoleaveofftheenclosingbracketswhenwritingdowna sequence:ifyouwritejust n2 ,yourreaderwillthinkthatyoumeanonlythe singlenumber n2 (forsomeparticular n thatyouhaveinmind)ratherthanthe wholeendlesslistofthesquares.
•Therearesomeoccasionswhen n = 1isnotthebeststartingpointfora sequence.If,forinstance,weneedtodiscussthesequence
n 1)(n 3) , thenwedarenotuse n = 1or n = 3becauseitwouldleadtodivisionbyzero (whichis,ofcourse,meaningless).Thenotationcanbetweakedslightlyto avoidthis,forexample,bywriting
whichstartsthelistoffsafelyat n = 4.
•Again,ifwewanttoworkwiththeendlesslistoffactorials,itmaybeusefulto recallthatzero-factorialisaperfectlygoodandusefulnumber,andexplicitlyto includeitinourdiscussionbyusinganotationsuchas
(n!)n≥0 .
Hereareafewillustrativeexamplesofsequences,somepresentedinmorethan onestyleofsymbol.Youmayfinditusefulto‘translatethemintoEnglish’inyour head;forinstance,thefirstis‘thesequenceofoddpositiveintegers’,thefourthis ‘thesequenceofprimes’,thesixthis‘thesequenceofreciprocalsofthepositive integersbutwiththesignalternating’andsoon.
2.2.1 Example 1. (1,3,5,7,9, ··· ) = (2n 1)n≥1 2. 3 2 , 3 4 , 9 8 , 15 16 , 33 32 , 63 64 , ··· = 1 + 1 2 ,1 1 22 ,1 +
3. 5, 1 2 ,5, 1 4 ,5, 1 8 ,5, 1 16 ,5, 1 32 , ··· = (xn ) where xn = 5if n isoddbut xn = 2 n/2 if n iseven.
4. (2,3,5,7,11, ··· ) = (yn )n≥1 where yn isthe nth primenumber.Noticehow potentiallymisleadingthefirstsymbolwashere:itcouldhavemeantseveral differentsequencesincluding,forexample,‘two,andthentheoddintegers excludingtheperfectsquares’.Thesecondsymbolwasfreefromanysuch ambiguity.
5. (1, 8,27, 64,125, 216, ) = (1, 8,27, 64, , ( 1)n 1 n 3 , ) = (( 1)n
Onceagainthefirstsymbolmighthavebeenmisunderstood,butthesecond andthirdleftnoroomforconfusion.
6.
7. (1, √2,3
Youshouldnoticethatsomesequences,butbynomeansallofthem,seemto besettlingtowardsan‘equilibriumvalue’,a‘steadystate’aswescanfurtherand furtheralongthelist.Forinstance,(2)aboveappearstobesettlingtowards1,and (6)towards0;incontrast,(1)and(4)aresofarshowingnosignofsettling,butare ‘explodingtowardsinfinity’(andofcourseweshallneedtomakethatphrasealot moreprecisebeforewedoanythingseriouswithit)while(5)isdoingsomekind ofcosmicsplitsbyexplodingtowardsinfinityandminusinfinityatthesametime (samecomment).Inthecaseofthelastfoursequences(7)to(10),itismuchless clear–tounaidedcommonsense–whatisgoingtohappeninthelongrun.
