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maticsshouldbeeitherprecededorfollowedbyanon-trivialexample,
andwedidourbesttoputourmoneywhereourmouthis.Another
standardplaceswantinganexamplearetheoremssayingthatflev-
ery
A
isa
B
”–theyshouldbefollowedeitherbyanexampleshowing
thatsome
B
’sarenot
A
’s,oraclearconfessionthattheauthorsdonot
knowwhethertheconversetowhattheyhaveprovedholds.While
mostcalculustextbooksfollowthattradition,booksonmoreadvanced
topics(liketheoneswedealwith)breakitalltoooften.Thereasonis
obvious–suchexamplesaresometimesverydifficulttocomeby.We
triedreallyhardtoprovidethemeverywherewecould.Sometimes
thatmeantusingclassicalones,butsometimesthatmeantfindingvery
obscureonesinsomelesser-knownpapers,andsometimesthatmeant
devisingthemourselves.(Wedidnotperformactualcalculations,but
webelievethattheresultisahigherthanaverageconcentrationof
examplesinourbook,andsomeofthemareindeedpublishedhere
forthefirsttime.)
Anothermeansweemployedinordertomakeotherwisetough
ordrypartsarejokes,popculturereferencesandgeneralgoofiness.
Wesincerelyhopethereaderwillenjoyfindinglittletidbitsoffunas
muchasweenjoyedhidingtheminvariousplaces.(Thisseemsoneof
thereasonsteachingmathsissuchawonderfulpastime–itmightbe
ratherawkwardtocracksillyjokesinabookon,say,militaryhistory.)
Thechoiceoftopicsmightseemsomewhatrandom.Infact,itis
farfromthat,althoughthereasonfortheseparticularthemesis,let
ussay,abitpersonal.Wedecidedtodiscussthingswhichweactually
knowsomethingabout,achoicethatwegenerallyrecommendboth
inacademiaandinlife.Thatdoesnotmeanthatthetopicsdonot
belongtogether,evenifthetiesbetweenthechaptersunderstoodas
flmostproofsfromchapter
(n+
1
)
dependontheresultsofchapter
n
”
areattimesabitloose.Infact,mostchapters(especiallyafterthe
oneonfixedpointtheory)canbereadalmostinanyorder.However,
abeginnermightbenefitmorefromstudyingthebookintheorderwe
wroteit.Onereasonisthatwesometimesdorelyonresultsknown
fromearlierchapters;anotheroneisthatinlaterchapterswearemore
succinct,assumingthatanyreaderwhogotthatfarmustbecompetent
I
