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7
ofthisbookwepreferredtoproveasimplerspecialcasethantoskip
theproofaltogether.(Well,flhard”beingsometimesdifficulttoassess,
ourcriterionwasreallymorelikeflletusproveasimplercaseifwe
wouldneedtoaddadozenofpagesandafewotherwiseunrelated
definitionsinstead”.Whiledigressionsarenicewhentellingfamily
stories–orinabookintroduction,forthatmatter–wepreferredto
makethestorylessdetailedthantoderailthetrainofthought.)
Whilenotourmaingoal,wehopethatonesideeffectofthelevelof
detailwedecidedtoincludecanbeteachingbuddingmathematicians
aboutthewaysofwritingproofs.Itisnotaneasytasktowritedown
aproofinaformthatisatthesametimepreciseandreadable,and
wespentmanyhourstryingtostrikethatbalance.Wealsostrived
tobeconsistentinnotation,andmakeitashelpfulaspossible.(Paul
Halmosoncefamouslywroteaboutflamathematician’snightmare–
asequence
nε
thattendsto0as
8
becomesinfinite”.Thisiseasyto
laughat,butinproofsusingalotofsequences,functions,setsand
whatnot,suchawkwardnesscanariseprettyinadvertently.)
Itdefinitelyneedstobeemphasizedthatweviewthisbookas
apossiblestartingpointtofurtherstudies.Ofcourse,somestudents
willuseittobothstartandfinishtheirnonlinearadventure(hopefully,
theywillatleastgetpastthisIntroduction...).Thatisperfectlyfine,
andwetriedtocaterforthembymakingthebookasself-contained
aspossible(sothatyoucouldtakeitwithyouonasix-monthtripto
Antarcticaandspendusefultimewithitevennothavingaccessto
alibraryortheInternet–andgivenitsweight,youcouldeventryto
killyouradvisorwithitiftheThinghappenedtoassimilatethem).
Ontheotherhand,wedosuggestreadingmore–manyapartofour
bookisreallyanappetizer,andinterestingstuffawaitstheintrepid
studentwhodecidestodigdeeper.Afterperusingourbook,readers
hungryformoreshouldbewellequippedtostudymoreadvanced
texts(betheyotherbooksorevenresearchpapers).
Attheendofeachchapter,weincludedaNotesandremarkssec-
tion.Theseservetwopurposes.Firstly,theygivepointerstoplaces
wherethingswetreatedmoresuperficiallyareexpanded.Westrongly
encouragethereaderstofollowthebreadcrumbtrailweleftthere.
I
