c# pdf free : Adding hyperlinks to pdf Library SDK class asp.net .net wpf ajax TRENCH_REAL_ANALYSIS1-part226

Section1.1
TheRealNumberSystem
3
Theorem1.1.1(TheTriangleInequality)
Ifaandbareanytworealnumbers;
then
jaCbjjajCjbj:
(1.1.3)
Proof
Therearefourpossibilities:
(a)
Ifa0andb0,thenaCb0,sojaCbjDaCbDjajCjbj.
(b)
Ifa0andb0,thenaCb0,sojaCbjDaC.b/DjajCjbj.
(c)
Ifa0andb0,thenaCbDjajjbj.
(d)
Ifa0andb0,thenaCbDjajCjbj.
Eq.1.1.3holdsincases
(c)
and
(d)
,since
jaCbjD
(
jajjbj ifjajjbj;
jbjjaj ifjbjjaj:
Thetriangleinequalityappearsinvariousformsinmanycontexts.Itisthemostimpor-
tantinequalityinmathematics.Wewilluseitoften.
Corollary1.1.2
Ifaandbareanytworealnumbers;then
jabj
ˇ
ˇ
jajjbj
ˇ
ˇ
(1.1.4)
and
jaCbj
ˇ
ˇ
jajjbj
ˇ
ˇ
:
(1.1.5)
Proof
Replacingabyabin(1.1.3)yields
jajjabjCjbj;
so
jabjjajjbj:
(1.1.6)
Interchangingaandbhereyields
jbajjbjjaj;
whichisequivalentto
jabjjbjjaj;
(1.1.7)
sincejbajDjabj.Since
ˇ
ˇ
jajjbj
ˇ
ˇ
D
(
jajjbj if jaj>jbj;
jbjjaj if jbj>jaj;
(1.1.6)and(1.1.7)imply(1.1.4).Replacingbbybin(1.1.4)yields(1.1.5),sincejbjD
jbj.
SupremumofaSet
AsetS ofrealnumbersisboundedaboveifthereisarealnumberbsuchthatx x   b
wheneverx 2 S. . Inthiscase,bisanupperboundofS. . IfbisanupperboundofS,
thensoisanylargernumber,becauseofproperty
(G)
.IfˇisanupperboundofS,butno
numberlessthanˇis,thenˇisasupremumofS,andwewrite
ˇDsupS:
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4 Chapter1
TheRealNumbers
Withtherealnumbersassociatedintheusualwaywiththepointsonaline,thesedefini-
tionscanbeinterpretedgeometricallyasfollows:bisanupperboundofSifnopointofS
istotherightofb;ˇDsupSifnopointofSistotherightofˇ,butthereisatleastone
pointofStotherightofanynumberlessthanˇ(Figure1.1.1).
(S = dark line segments)
β
b
Figure1.1.1
Example1.1.1
IfSisthesetofnegativenumbers,thenanynonnegativenumberisan
upperboundofS,andsupSD0.IfS
1
isthesetofnegativeintegers,thenanynumbera
suchthata1isanupperboundofS
1
,andsupS
1
D1.
Thisexampleshowsthatasupremumofasetmayormaynotbeintheset,sinceS
1
containsitssupremum,butSdoesnot.
Anonemptysetisasetthathasatleastonemember.Theemptyset,denotedby;,isthe
setthathasnomembers.Althoughitmayseemfoolishtospeakofsuchaset,wewillsee
thatitisausefulidea.
TheCompleteness Axiom
Itisonethingtodefineanobjectandanothertoshowthattherereallyisanobjectthat
satisfiesthedefinition. (Forexample, , doesitmakesensetodefinethesmallestpositive
realnumber?)Thisobservationisparticularlyappropriateinconnectionwiththedefinition
ofthesupremumofaset. Forexample,theemptysetisboundedabovebyeveryreal
number, soithasnosupremum. . (Thinkaboutthis.) Moreimportantly,wewillseein
Example1.1.2thatproperties
(A)
(H)
donotguaranteethateverynonemptysetthat
isboundedabovehasasupremum. Sincethispropertyisindispensabletotherigorous
developmentofcalculus,wetakeitasanaxiomfortherealnumbers.
(I)
Ifanonemptysetofrealnumbersisboundedabove,thenithasasupremum.
Property
(I)
iscalledcompleteness,andwesaythattherealnumbersystemisacomplete
orderedfield.Itcanbeshownthattherealnumbersystemisessentiallytheonlycomplete
orderedfield; thatis, , ifanalienfrom anotherplanetweretoconstructamathematical
systemwithproperties
(A)
(I)
,thealien’ssystemwoulddifferfromtherealnumber
systemonlyinthatthealienmightusedifferentsymbolsfortherealnumbersandC,,
and<.
Theorem1.1.3
IfanonemptysetSofrealnumbersisboundedabove;thensupSis
theuniquerealnumberˇsuchthat
(a)
xˇforallxinSI
(b)
if>0.nomatterhowsmall/;thereisanx
0
inSsuchthatx
0
>ˇ:
Section1.1
TheRealNumberSystem
5
Proof
WefirstshowthatˇDsupS hasproperties
(a)
and
(b)
. Sinceˇisanupper
boundofS,itmustsatisfy
(a)
.Sinceanyrealnumberalessthanˇcanbewrittenasˇ
withDˇa>0,
(b)
isjustanotherwayofsayingthatnonumberlessthanˇisan
upperboundofS.Hence,ˇDsupSsatisfies
(a)
and
(b)
.
Nowweshowthattherecannotbemorethanonerealnumberwithproperties
(a)
and
(b)
. Supposethatˇ
1
2
andˇ
2
hasproperty
(b)
;thus,if>0,thereisanx
0
inS
suchthatx
0
2
.Then,bytakingDˇ
2
ˇ
1
,weseethatthereisanx
0
inSsuch
that
x
0
2
.ˇ
2
ˇ
1
/Dˇ
1
;
soˇ
1
cannothaveproperty
(a)
. Therefore,therecannotbemorethanonerealnumber
thatsatisfiesboth
(a)
and
(b)
.
SomeNotation
WewilloftendefineasetS bywritingS D
˚
x
ˇ
ˇ

,whichmeansthatSconsistsofall
xthatsatisfytheconditionstotherightoftheverticalbar;thus,inExample1.1.1,
SD
˚
x
ˇ
ˇ
x<0
(1.1.8)
and
S
1
D
˚
x
ˇ
ˇ
xisanegativeinteger
:
Wewillsometimesabbreviate“xisamemberofS”byx2S,and“xisnotamemberof
S”byx…S.Forexample,ifSisdefinedby(1.1.8),then
12S but 0…S:
TheArchimedeanProperty
ThepropertyoftherealnumbersdescribedinthenexttheoremiscalledtheArchimedean
property. Intuitively,itstatesthatitispossibletoexceedanypositivenumber,nomatter
howlarge,byaddinganarbitrarypositivenumber,nomatterhowsmall,toitselfsufficiently
manytimes.
Theorem1.1.4(ArchimedeanProperty)
Ifandarepositive;thenn>
forsomeintegern:
Proof
Theproofisbycontradiction. Ifthestatementisfalse,isanupperboundof
theset
SD
˚
x
ˇ
ˇ
xDn;nisaninteger
:
Therefore,Shasasupremumˇ,byproperty
(I)
.Therefore,
nˇ forallintegersn:
(1.1.9)
6 Chapter1
TheRealNumbers
SincenC1isanintegerwhenevernis,(1.1.9)impliesthat
.nC1/ˇ
andtherefore
nˇ
forallintegersn.Hence,ˇisanupperboundofS.Sinceˇ<ˇ,thiscontradicts
thedefinitionofˇ.
DensityoftheRationalsandIrrationals
Definition1.1.5
AsetDisdenseintherealsifeveryopeninterval.a;b/containsa
memberofD.
Theorem1.1.6
TherationalnumbersaredenseintherealsIthatis,ifaandbare
realnumberswitha<b;thereisarationalnumberp=qsuchthata<p=q<b.
Proof
FromTheorem1.1.4withD1andDba,thereisapositiveintegerqsuch
thatq.ba/>1.Thereisalsoanintegerj suchthatj j >qa.Thisisobviousifa0,
anditfollowsfromTheorem1.1.4withD1andDqaifa>0.Letpbethesmallest
integersuchthatp>qa.Thenp1qa,so
qa<pqaC1:
Since1<q.ba/,thisimpliesthat
qa<p<qaCq.ba/Dqb;
soqa<p<qb.Therefore,a<p=q<b.
Example1.1.2
Therationalnumbersystemisnotcomplete;thatis,asetofrational
numbersmaybeboundedabove(byrationals),butnothavearationalupperboundless
thananyotherrationalupperbound.Toseethis,let
SD
˚
r
ˇ
ˇ
risrationalandr
2
<2
:
Ifr2S,thenr<
p
2.Theorem1.1.6impliesthatif>0thereisarationalnumberr
0
suchthat
p
2<r
0
<
p
2,soTheorem1.1.3impliesthat
p
2DsupS.However,
p
2is
irrational;thatis,itcannotbewrittenastheratioofintegers(Exercise1.1.3). Therefore,
ifr
1
isanyrationalupperboundofS,then
p
2<r
1
.ByTheorem1.1.6,thereisarational
numberr
2
suchthat
p
2<r
2
<r
1
.Sincer
2
isalsoarationalupperboundofS,thisshows
thatShasnorationalsupremum.
Sincetherationalnumbershaveproperties
(A)
(H)
, butnot
(I)
,thisexampleshows
that
(I)
doesnotfollowfrom
(A)
(H)
.
Theorem1.1.7
ThesetofirrationalnumbersisdenseintherealsIthatis,ifaandb
arerealnumberswitha<b;thereisanirrationalnumbertsuchthata<t<b:
Section1.1
TheRealNumberSystem
7
Proof
FromTheorem1.1.6,therearerationalnumbersr
1
andr
2
suchthat
a<r
1
<r
2
<b:
(1.1.10)
Let
tDr
1
C
1
p
2
.r
2
r
1
/:
Thentisirrational(why?)andr
1
<t<r
2
,soa<t<b,from(1.1.10).
InfimumofaSet
AsetS ofrealnumbersisboundedbelowifthereisarealnumberasuchthatx x  a
wheneverx 2S. Inthiscase,aisalowerboundofS. IfaisalowerboundofS,sois
anysmallernumber,becauseofproperty
(G)
.If˛isalowerboundofS,butnonumber
greaterthan˛is,then˛isaninfimumofS,andwewrite
˛DinfS:
Geometrically,thismeansthattherearenopointsofStotheleftof˛,butthereisatleast
onepointofStotheleftofanynumbergreaterthan˛.
Theorem1.1.8
IfanonemptysetSofrealnumbersisboundedbelow;theninfSis
theuniquerealnumber˛suchthat
(a)
x˛forallxinSI
(b)
if>0.nomatterhowsmall/,thereisanx
0
inSsuchthatx
0
<˛C:
Proof
(Exercise1.1.6)
AsetSisboundediftherearenumbersaandbsuchthataxbforallxinS.A
boundednonemptysethasauniquesupremumandauniqueinfimum,and
infSsupS
(1.1.11)
(Exercise1.1.7).
TheExtendedRealNumberSystem
AnonemptysetS ofrealnumbersisunboundedaboveifithasnoupperbound,orun-
boundedbelowifithas nolowerbound. . Itisconvenienttoadjointotherealnumber
systemtwofictitiouspoints,C1(whichweusuallywritemoresimplyas1)and1,
andtodefinetheorderrelationshipsbetweenthemandanyrealnumberxby
1<x<1:
(1.1.12)
Wecall1and1pointsatinfinity.IfSisanonemptysetofreals,wewrite
supSD1
(1.1.13)
toindicatethatSisunboundedabove,and
infSD1
(1.1.14)
toindicatethatSisunboundedbelow.
8 Chapter1
TheRealNumbers
Example1.1.3
If
SD
˚
x
ˇ
ˇ
x<2
;
thensupSD2andinfSD1.If
SD
˚
x
ˇ
ˇ
x2
;
thensupS D D 1andinfS S D D 2. . IfS S isthesetofallintegers, thensupS S D D 1and
infSD1.
Therealnumbersystemwith1and1adjoinediscalledtheextendedrealnumber
system,orsimplytheextendedreals.Amemberoftheextendedrealsdifferingfrom1
and1isfinite;thatis,anordinaryrealnumberisfinite. However, , theword“finite”in
“finiterealnumber”isredundantandusedonlyforemphasis,sincewewouldneverrefer
to1or1asrealnumbers.
Thearithmeticrelationshipsamong1,1,andtherealnumbersaredefinedasfollows.
(a)
Ifaisanyrealnumber,then
aC1D
1CaD
1;
a1D1CaD1;
a
1
D
a
1
D0:
(b)
Ifa>0,then
a1
D
1a D
1;
a.1/D.1/aD1:
(c)
Ifa<0,then
a1
D
1a D1;
a.1/D.1/aD
1:
Wealsodefine
1C1D11D.1/.1/D1
and
11D1.1/D.1/1D1:
Finally,wedefine
j1jDj1jD1:
Theintroductionof1and1,alongwiththearithmeticandorderrelationshipsdefined
above,leadstosimplificationsinthestatementsoftheorems. Forexample,theinequality
(1.1.11),firststatedonlyforboundedsets,holdsforanynonemptysetSifitisinterpreted
properlyinaccordancewith(1.1.12)andthedefinitionsof(1.1.13)and(1.1.14). Exer-
cises1.1.10
(b)
and1.1.11
(b)
illustratetheconvenienceaffordedbysomeofthearith-
meticrelationshipswithextendedreals,andotherexampleswillillustratethisfurtherin
subsequentsections.
Section1.1
TheRealNumberSystem
9
Itisnotusefultodefine11,01,1=1,and0=0.Theyarecalledindeterminate
forms,andleftundefined. Youprobablystudiedindeterminateformsincalculus;wewill
lookatthemmorecarefullyinSection2.4.
1.1Exercises
1.
Writethefollowingexpressionsinequivalentformsnotinvolvingabsolutevalues.
(a)
aCbCjabj
(b)
aCbjabj
(c)
aCbC2cCjabjC
ˇ
ˇ
aCb2cCjabj
ˇ
ˇ
(d)
aCbC2cjabj
ˇ
ˇ
aCb2cjabj
ˇ
ˇ
2.
Verifythatthesetconsistingoftwomembers,0and1,withoperationsdefinedby
Eqns.(1.1.1)and(1.1.2),isafield.Thenshowthatitisimpossibletodefineanorder
<onthisfieldthathasproperties
(F)
,
(G)
,and
(H)
.
3.
Showthat
p
2isirrational. H
INT
:Showthatif
p
2D m=n;wheremandnare
integers;thenbothmandnmustbeeven:Obtainacontradictionfromthis:
4.
Showthat
p
pisirrationalifpisprime.
5.
FindthesupremumandinfimumofeachS.StatewhethertheyareinS.
(a)
SD
˚
x
ˇ
ˇ
xD.1=n/CŒ1C.1/nn2;n1
(b)
SD
˚
x
ˇ
ˇ
x
2
<9
(c)
SD
˚
x
ˇ
ˇ
x7
(d)
SD
˚
x
ˇ
ˇ
j2xC1j<5
(e)
S D
˚
x
ˇ
ˇ
.x
2
C1/
1
>
1
2
(f)
SD
˚
x
ˇ
ˇ
xDrationalandx7
6.
ProveTheorem1.1.8. H
INT
:ThesetT D
˚
x
ˇ
ˇ
x2S
isboundedaboveifSis
boundedbelow:Applyproperty
(I)
andTheorem1.1.3toT:
7. (a)
Showthat
infSsupS
.A/
foranynonemptysetS ofrealnumbers, andgivenecessaryandsufficient
conditionsforequality.
(b)
ShowthatifS isunboundedthen(A)holdsifitisinterpretedaccordingto
Eqn.(1.1.12)andthedefinitionsofEqns.(1.1.13)and(1.1.14).
8.
LetSandT benonemptysetsofrealnumberssuchthateveryrealnumberisinS
orTandifs2Sandt2T,thens<t.Provethatthereisauniquerealnumberˇ
suchthateveryrealnumberlessthanˇisinSandeveryrealnumbergreaterthan
ˇisinT. (Adecompositionoftherealsintotwosetswiththesepropertiesisa
Dedekindcut.ThisisknownasDedekind’stheorem.)
10 Chapter1
TheRealNumbers
9.
Usingproperties
(A)
(H)
oftherealnumbersandtakingDedekind’stheorem
(Exercise1.1.8)asgiven,showthateverynonemptysetU ofrealnumbersthatis
boundedabovehasasupremum.H
INT
: LetTbethesetofupperboundsofU U and
SbethesetofrealnumbersthatarenotupperboundsofU:
10.
LetSandT benonemptysetsofrealnumbersanddefine
SCT D
˚
sCt
ˇ
ˇ
s2S;t2T
:
(a)
Showthat
sup.SCT/DsupSCsupT
.A/
ifSandTareboundedaboveand
inf.SCT/DinfSCinfT
.B/
ifSandTareboundedbelow.
(b)
Showthatiftheyareproperlyinterpretedintheextendedreals,then(A)and
(B)holdifSandTarearbitrarynonemptysetsofrealnumbers.
11.
LetSandT benonemptysetsofrealnumbersanddefine
ST D
˚
st
ˇ
ˇ
s2S;t2T
:
(a)
ShowthatifSandT arebounded,then
sup.ST/DsupSinfT
.A/
and
inf.ST/DinfSsupT:
.B/
(b)
Showthatiftheyareproperlyinterpretedintheextendedreals,then(A)and
(B)holdifSandTarearbitrarynonemptysetsofrealnumbers.
12.
LetS beaboundednonemptysetofrealnumbers,andletaandbbefixedreal
numbers.DefineT D
˚
asCb
ˇ
ˇ
s2S
.FindformulasforsupTandinfTinterms
ofsupSandinfS.Proveyourformulas.
1.2MATHEMATICALINDUCTION
Ifaflightofstairsisdesignedsothatfallingoffanystepinevitablyleadstofallingoffthe
next,thenfallingoffthefirststepisasurewaytoendupatthebottom.Crudelyexpressed,
thisistheessenceoftheprincipleofmathematicalinduction:Ifthetruthofastatement
dependingonagivenintegernimpliesthetruthofthecorrespondingstatementwithn
replacedbynC1,thenthestatementistrueforallpositiveintegersnifitistruefornD1.
Althoughyouhaveprobablystudiedthisprinciplebefore,itissoimportantthatitmerits
carefulreviewhere.
Peano’sPostulatesandInduction
TherigorousconstructionoftherealnumbersystemstartswithasetNofundefinedele-
mentscallednaturalnumbers,withthefollowingproperties.
Section1.2
MathematicalInduction
11
(A)
Nisnonempty.
(B)
Associatedwitheachnaturalnumbernthereisauniquenaturalnumbern
0
called
thesuccessorofn.
(C)
Thereisanaturalnumber
nthatisnotthesuccessorofanynaturalnumber.
(D)
Distinctnaturalnumbershavedistinctsuccessors;thatis,ifn¤m,then n
0
¤m
0
.
(E)
TheonlysubsetofNthatcontains
nandthesuccessors ofallitselementsis N
itself.
TheseaxiomsareknownasPeano’spostulates. Therealnumberscanbeconstructed
fromthenaturalnumbersbydefinitionsandargumentsbasedonthem.Thisisaformidable
taskthatwewillnotundertake.Wementionittoshowhowlittleyouneedtostartwithto
constructtherealsand,moreimportant,todrawattentiontopostulate
(E)
,whichisthe
basisfortheprincipleofmathematicalinduction.
ItcanbeshownthatthepositiveintegersformasubsetoftherealsthatsatisfiesPeano’s
postulates(with
nD1andn
0
DnC1),anditiscustomarytoregardthepositiveintegers
andthenaturalnumbersasidentical.Fromthispointofview,theprincipleofmathematical
inductionisbasicallyarestatementofpostulate
(E)
.
Theorem1.2.1(PrincipleofMathematicalInduction)
LetP
1
;P
2
;...;
P
n
;... bepropositions;oneforeachpositiveinteger;suchthat
(a)
P
1
istrueI
(b)
foreachpositiveintegern;P
n
impliesP
nC1
:
ThenP
n
istrueforeachpositiveintegern:
Proof
Let
MD
˚
n
ˇ
ˇ
n2NandP
n
istrue
:
From
(a)
,12M,andfrom
(b)
,nC12Mwhenevern2M. Therefore,MDN,by
postulate
(E)
.
Example1.2.1
LetP
n
bethepropositionthat
1C2CCnD
n.nC1/
2
:
(1.2.1)
ThenP
1
isthepropositionthat1D 1,whichiscertainlytrue. . IfP
n
istrue,thenadding
nC1tobothsidesof(1.2.1)yields
.1C2CCn/C.nC1/D
n.nC1/
2
C.nC1/
D.nC1/
n
2
C1
D
.nC1/.nC2/
2
;
or
1C2CC.nC1/D
.nC1/.nC2/
2
;
12 Chapter1
TheRealNumbers
whichisP
nC1
,sinceithastheformof(1.2.1),withnreplacedbynC1.Hence,P
n
implies
P
nC1
,so(1.2.1)istrueforalln,byTheorem1.2.1.
Aproofbased onTheorem 1.2.1 is aninductionproof, orproofbyinduction. . The
assumptionthatP
n
istrueistheinductionassumption.(Theorem1.2.3permitsakindof
inductionproofinwhichtheinductionassumptiontakesadifferentform.)
Induction,bydefinition,canbeusedonlytoverifyresultsconjecturedbyothermeans.
Thus,inExample1.2.1wedidnotuseinductiontofindthesum
s
n
D1C2CCnI
(1.2.2)
rather,weverifiedthat
s
n
D
n.nC1/
2
:
(1.2.3)
Howyouguesswhattoprovebyinductiondependsontheproblemandyourapproachto
it.Forexample,(1.2.3)mightbeconjecturedafterobservingthat
s
1
D1D
12
2
; s
2
D3D
23
2
; s
3
D6D
43
2
:
However, thisrequiressufficientinsighttorecognizethattheseresultsareoftheform
(1.2.3)fornD1,2,and3. Althoughitiseasytoprove(1.2.3)byinductiononceithas
beenconjectured,inductionisnotthemostefficientwaytofinds
n
,whichcanbeobtained
quicklybyrewriting(1.2.2)as
s
n
DnC.n1/CC1
andaddingthisto(1.2.2)toobtain
2s
n
DŒnC1CŒ.n1/C2CCŒ1Cn:
Therearenbracketedexpressionsontheright,andthetermsineachadduptonC1;
hence,
2s
n
Dn.nC1/;
whichyields(1.2.3).
Thenexttwoexamplesdealwithproblemsforwhichinductionisanaturalandefficient
methodofsolution.
Example1.2.2
Leta
1
D1and
a
nC1
D
1
nC1
a
n
; n1
(1.2.4)
(wesaythata
n
isdefinedinductively),andsupposethatwewishtofindanexplicitformula
fora
n
.ByconsideringnD1,2,and3,wefindthat
a
1
D
1
1
; a
2
D
1
12
; and a
3
D
1
123
;
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