Real Analysis: A Historical Approach

316 Pages·2011·3.76 MB·English

by  Saul Stahl

2 Puraen Adp plMiaetdh ematics:
A WilSeeriesy ofTeMlotnso,g raanTpdrh asc,t s
Real
Analysis
A HistoriAcpaplr oach
SI.(()\D IDIII()\
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ReaAln alysis
PUREA NDA PPLIEMDA THEMATICS
A WileSye rioefsT extMso,n ograpahnsdT, racts
FoundebdyR ICHARDC OURANT
EditoErmse ritMiY:R ON ALLEN IllD,A VIDA .C OX,P ETERH ILTON,
B.
HARRY HOCHSTADTP,E TERL AX,J OHNT OLAND
A compleltieso tft het itlients h isse riaepsp eaartts h ee ndo ft hivso lume.
ReaAln alysis:
A HistorAipcparlo ach
SeconEdd ition
SauSlt ahl
TheU niverosfiKt ayn sas
DepartmeonftM athematics
LawernceK,S
�WILEY
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ISBN9 78-0-470-87(8h9a0r-d3b ack)
I.M athemataincaally s2i.Fs u.n ctioofnr se avla riabIl.eT si.t le.
QA300.S8280211
515'.8-dc22 2011010976
Printientd h eU niteSdt atoefsA merica.
oBooIkS BN:9 78118096864
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ePuIbS BN:9 78118096857
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Contents
Prefatcote h See coEnddi tion xi
Acknloewdgments xv
1 Archiemsea dndt hPea rabola 1
1.1 TheA reoaf t hPea rabolSiecg ment 1
1.2*T heG eometorfty h Pea rabola 8
2 FermatD,i fferentiaantdIi notne,g ration 13
2.1F erma'tsC alculus 13
3 NewtonC'asl culus1 )( Part 19
3.1 TheF ractioBnianlo miTahle orem 19
3.2A reasa ndI nfiniSteer ies 23
3.3N ewtoPnro'osf s 30
4 NewtonC'asl cu(lPuasr t 2) 35
4·1 TheS olutoifDo inff enretiEaqlau tions 35
4·2*T heS olutoifAo lng eibcEra qautions 40
ChatpeArp pendMiaxt:h emaItimcpal ementoaft ions
NewtoAnl'osgr ithm 48
vii
viiCiO TNENTS
5 Euler 51
5.1 TrigonomSeetrriiecs 51
6 TheR eaNlu mbers 61
6.1A nI nforImnatlro duction 61
6.2 OrderFeiedl ds 64
6.3 CompleteannedsI srr atiNounmable rs 71
6.*4 TheE uclidPeroacens s 76
6.5F unctions 80
7 SequenacnedTs h eiLri mits 85
7.1 TheD efinitions 85
7.2L imit Theorems 92
8 TheC auchPyro perty 103
8.1L imiotfsM onotoSneeq uences 103
8.2 TheC auchPyro perty 111
9 TheC onverego efIn ncfiniSteer ies 115
9.1S tocSke ries 115
9.2S erioefPs o sitiTveer ms 120
9.3S erioefAs r bitraTreyr ms 124
9.*4 TheM osCte lebraPtroebdl em 132
10S erioefFs u nctions 139
10.P1o weSre ries 139
10.2T rigonomSeetrriiecs 145
11C ontinuity 149
11.A1n I nforImnatlro duction 149
11.2 Theo fLa iF muintc tion 150
11.3C ontinuity 155
11.4P ropertoifCe osn tinuFouunsc tions 163
12D ifferentiability 169
12.A1n I nforImnatlro ducttoDi ioffne rentiati1o6n9
12.2T heD erivative 171
12.3T heC onesquenocfeD si fferentiability1 77