Methods of Real Analysis(Richard R. Goldberg)

410 Pages·2016·24.2 MB·English

RICHARD R.
GOLDBERG

The Uae
of lowe

SECOND FOITION
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JOHN YALEY & SONS, In.
GSTS Ge nd com Reser Capa)

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PREFACE

ny book ie inseuded us x ouryees couree for sludems wbo ‘uve cenipleld ea
nlinusy sequence of cours im eleaeriry culls I presents ia rigoreus fashion basic
rmzseial om the fundurmeatul concep ual “ools of aralyse—functons, lil, coatinu=
ity dervather usd integrals, sequences ung seres, Most of -e dita ports aly
lovin uver an elementary course: ace deli ooh iz dela, well ut many mere
rulegneed sop desinell lo gee w gout backround Zor fuze hopefully use of)
rederu wnulis and topology. 1a paroculu, there ure weattoents of meer sputes und
Tebergne calearution, Wyice that are ule eeserved for mere wlvamesd course, Alo
inluted are racay swaller bur interesting ties not esuly peeented 3 courses ut bs
level the taies inclute Baie eatery aad Jseoutinvae Fometaors, semmablly of
sefis, the Welostrase thgorein cer appiosinaige of ccauous fvnetions by pl
bowials, end a pono! ofthe sandard existense theorem ne allan! eau Eoom
Ue paint of wot af Fxed-poie theory

te book ie wien at the same level as texts for cadiional “advaced calcul”
course, bul dot not cousider opie in "soveralverables.” Marerial on cffereule and
cor cali, a eur upimin, cea be unders.oo best from ike ant uF view ul ee
Aiverental yore el belong 2 & pare ere

REMARKS ON THE SECOND EDITION.

Macy daanges in, niiinas tal ene Jeleuwe (wnt J ft aion have been
made in aceon wil Magnghi(l extn fen ears callesqoes alge wel al

‘A naajot festoce oF SHG new edition in dhe saline 1 seine called “Noles aed
Adgitonal Exerciecs” sich include vasely nf materiel, Tene ae fama these
related co the material 20 the body of the txt” for example, the SehvOder-Ravnacia
Ueoren fom set dheary, tbe Tetas exteasion cheorsa from ropelagy, and Stone's
generalztion oF the Weie-sease approximation chcore3. Tho prac ave given :o ontive
Fad with reat del ett the salen as exerci. Un hove ew seen there ur Ao
myeluneeus exert anuay of whit are jute chelengitg) aud en eceasioual bist

2° now Ar insrtctor’ sollicns suarual tor the proslesns ia the new weal can be
sai fram the sath

alse ace aa eppembs thi. contac ex wave (ralment uf he seal amber
start Thie wax a enmprise belween ma estan lim the fre eign al 4
Tent evelopment Feenbasie principles Sat T rink would etal the readers propa
intn the ene of the book. The aun plone how, the re esbers an the necessary
feu that es be erie Pus these naan are are prevented

“Theve ane a nucber uf util ilugeevone a's 8 Separtne fron The Fe
and gsr exetitenoty of th cer gel nem ps

Ricbond & Gling
CONTENTS,

Iorodnor Assure and Notations

MM
12.
Ww.

2
22,
2a
2A
2s
28,
28
28.
2.10
2a

Sets and functions

Sets and eles
Oparoions on sae
eabvolcd fnstons
Fauivoence counbilty
eal numbers

ect upper bounds

Sequences of roal numbers

Defiin of sequence and subsequence
Lint of sesuence

Convergent seguances

Drstaont sequences

Bounded stqvonces

Monstane saquencos

Operaion on vergent sequencer
Lint super ond inferior

Couey sequencer

Surtmabilty of sequences
2212. lint superior and lr insiar tr anqueres af see

Series of real numbers

8.1. Convergence and divergence
2.2. Series with nonmegafie erm
2.3. Alleating sein
3.A. Conditional convergence and absolle convergence
2.5. Reerongemens of secien
118. Tet for belie conwergance
2.7. Sevos whase tems form 2 naninicoring sequence
28, Summation by pars
32. 16,3) sunmabity of eres
240. The cass?
3.11, Real mumbets end dasial expensions
2.02, Note and adele! nares for
(Chapters 1, 2, and 3

mits ond metre spacer
421, Lint ofa Fctlon oth el ie

4.2 Moe paces

Continuous funetions on metre spaces
5.1. Fantionscomiuous ata point an he rea ne
152. Reformation

453. Funlionscaninuovs on ami pace

54. Open sets

55. Clad ste

534. Diconinsow Rneons on A!

5.7. The lite fom point oa et

Connectedness, completenet, anu compactnoss
62. Camectd sa

(62. Bounded see and oly hounded st

0.4. Complete mei eacee

65. Compute etic ses

v8
149
“9
6.5. Contnueus functions on compoct metic paces 18
67. Conti of the avez freon 166
68. Uniform coninsty wer
69, Noles and adconal exercises for

Chapters 4, 5am ”
Calculus we
7A. Sets. moni 20 w
7.2, Defi ofthe Bemann inegrat 180
7.3. Exislence of fhe Riemann integral 16
7.4. Propeties ofthe Reman integral 18
75. Bervatves 9
7.8. Real's theorem 290
77, Thelav ot thereon 203
7.8, Fundamental theorems of elou ut 208
7.9. Improper negro a
7.10. improper integra continued 28

8. The elemontary functions. Tayler serios m
8.1. Hypirblc fans ms
8.2, The exponent onston ore
8.3, The logeitmic Kineton, Denton of e* poy
BA. The iganomeniefuncions 200
8.5, Toplrs theorem 28
8.46, The binomial theorem an
8.7. Hospital as

9. Sequences and series oF Functions ae
9.1. Pointe convergence of sequence of unctions a2
9.2. Uniform convergence of sequences at fncions 28
9.3. Consequences of uitoem convergence 0
9.4. Convergence and uniform convergence af sie af Rnctions 264
9.5. Iteqation and difrenition of ste of enchonn 268
9.6. Abe surablly m
9.7. 2 continuous, nowhere ditreatobe fonction a

10. Three famous theorems

16,1, The meine space Clas
10.2. Thy Waiartans approximation theorem 20
10, Peotdasiance shooter fr ciel equations or
10.4, The Arzlathetem on equicominuaus fancy 0

DLS. Notes and edciional exercises for

Chopters # and 10 2a

1, The Lebesgue integral m0
V1. Lonath of epan os anl lose sate 9
1.2, liner and puer means. Measurable = wen
11.5. Propenties of measuabie ste 208
114. Manabe fenefione m
11.5. finan and exnlence ofthe Lnberpue ing for bowl

functions as
LTS. Properties of he Lebesgue inogel for bowed
essere fn a

S12. The ebexgue infer or anbourddfnehions me
11. Sere fda Merete 28
11.9, The mete soe £0. oa
11.19, The ata! on (= 20} arin he plone on

12, Fourier corias 25
12.1. Definion of Fura ar as
12.2. Felon of soneigeee probes 358
12.8. The (C1) emmy of Fer serie 362
12st, The E? teary of Forres 38
12.5. Coneigene f Futer snes 368
12.6. Onhonomal axpensions I Ela. wa
27, Notes an ecltonal sxe for

‘Chapars 17 and 12 300,

Appendix: the algebraic and ower oso

for ha el rombae 8
Special Symbols 5
Index we
INTRODUCTION

ASSUMPTIONS
AND NOTATIONS

A. The book det ent agin with an entsnded sovelopmtn of the real rumen.
However, he reads: who wishes © proceed ia aviely Toye ler shoul Gr get the
tusie detiniions aad dhearotas aboot set and Funeione fa Seoiags [trough (3nd
her cum 18 che Appanduc for dhe slgeors and order axioms of the veat and “he
‘eorewn sr svihertte aml sequels that aoe derived trom thoee atome Aer “ae
Avert the reilr shoul gry Selion 1 ehere cht ceil epper aeune axiom ie
pickerel, AT this point the reater wil have seeo a saree ieraens of all the baste
Essunptions about the res'e Anon clinsng Fe appro ay sip mow ly paragraph
a

These are sine whe fel, ewer, thal HL petra ist be Le form.
hoot Ge roi! amar a Ih he mle ev ge eee mal oF the bovk mere Quy.
Fromm is ania of vicw i halter I daly naling (ie “Appencix ane simply Grover
Sireny throng Ge main hed of the est For thane shu ooh se take Ths uppronch we
arancon hretly same fasts aba the nels

"An integer is a"wale nua?” Thus 6, 3 ae eyes A rama neers re
umber dist can bo sxovstied as a quoient of itegere Thos 3/2 snl 9,378 one
‘ional acrobers. Ans integer £ Ig Cm & rational Mather singe we way write FAL
‘a uational unter is 3eal pumber that ix a 4 cainssl wowiher. For exaenne,
Soluion to the ection a2 muse be an aradonal uber,

‘The render shotl have some Eatity sn handling iroquaidies, He stould know the: if
sand 7 ate cod) autabers aa 0, thea —y. Alg, # Ushaeey. then Oe 15

For c50 we define 110 be Fin ¢0 we define [a to be —, Pally we deine
10]. Thus for ang veal numbers, [x 3 he numeral value” of We eal [sche
bsolute vale of «By considering varieus eases secouiny oh ga af «and whe
reader aould hase nn detienlty in penving the smsvensyimportaat TEES

betel el uy

and
Wo aad fh are eal nuabsrs, hen the geometie iacraretation of lx A] the dicance
rom a to b for from & 16 a) This ntetprctation ie cepceally mpparaat for the
Undevtaading of he ctental ideas: many of che proais. I a, «ate veal comers,
fe geemncri weaning ofthe inequaley

Jone sane k a
ie that the distance Hom w 10 b is aa grsezer than the distance from uta pls the
Aistanee frome e 40 This should seem quits reascnabe. Soe If you van prove (2). [Lee
sea eyre hand xs t}]

CW assume eis truth of laws af exponents auch ¢5 2°17 —a°Q? for aU aad Jor
vation valase of « and, 10 Chapter ow define tor cy real number » aad her
prove the amie vs ef exponent ir arhise-y exponents. The nating 2 and Yu
Fath cea the peiive seanae ol wo The axikonce a! # pave guar oor fas aay
restive sue ie putentad in euatine 82 Saatita 62)

1D. Ha aie sone npiatgcn, He and fe as scl mers wit a we dsaore Py
(ey the st al eal sere « ash that a Ry (2.25) wo mean the at ofall
ISal Sich that esa. Be fnseya) we mean these a all cal a Wiha a, The so 4F)
Jncallod a botnded opeafotsival while | sepa} ad [u-cel ate calcd unbounded open
fnvervas, The set of el real numer is soncetisnes denetee Ly { <2.) Nate that We
‘arent defining the symbol:

Taf. the [od] denoces tis set of ral namers 4 auch thac 2.4.74 5. These ie
‘alle bowaded closed bverval, A clored inerval ay ths contain ouly one post (it

1 We oscesonaly need to aie “bed-open” terete, Por example. 01) dese the
Interval of aumbere.© wich she 1

‘ear use ha station a6) ta denote pin a the pee, AB we mill os, Ae
point whase Me-eonediaate” is a and whose “y-scordinate™ is wil he donotee by
agh

is oftea cantoaioat crits in parontosss. tothe right ofa digslayed tome. the
values of ts variable” oo arabes? fc which the stateaten tte, For exam

fier awn

wens Cutt the imber fy) 3 Jess chao 7 er a» in [23h

'B Ihe mutenel io this buok is logically independien: of courses io eiemenssry
ecrnetsy, Iigononiery exe! eteule hat. me be ao esol roca these elementary
ourses Ht uy definigon or eu tbe stecemeat ur prest ef a theucent unlese ae have
previously eolublstiel tie resi ouelves, Neverbele, we une Greely resule a
Soncepty Ero emensiny elculy to lctere ou delinurs ail theorems, Vai, 23
fearaple, we de nol Ueline the siwefuuetion yes Caples 8, But we do we laniiar
‘colt aoa tbe sie fomctur.f exnmpey ama exgucse a elon chaptes

“Therese seme petri Tumainne onthe text. Dut nora gat many. We Delve thet
{Ge teadar thorldleacs 9 daar his wa pitures as carly a possible, Prestaably. the
tasteuecer wil help 2) fhe rough spo in eis ake,