c# pdf free : Add link to pdf file Library application component asp.net html web page mvc TRENCH_REAL_ANALYSIS2-part237

Section1.2
MathematicalInduction
13
andthereforeweconjecturethat
a
n
D
1
:
(1.2.5)
ThisisgivenfornD1.Ifweassumeitistrueforsomen,substitutingitinto(1.2.4)yields
a
nC1
D
1
nC1
1
D
1
.nC1/Š
;
whichis(1.2.5)withnreplacedbynC1. Therefore,(1.2.5)istrueforeverypositive
integern,byTheorem1.2.1.
Example1.2.3
Foreachnonnegativeintegern,letx
n
bearealnumberandsuppose
that
jx
nC1
x
n
jrjx
n
x
n1
j; n1;
(1.2.6)
whererisafixedpositivenumber.Byconsidering(1.2.6)fornD1,2,and3,wefindthat
jx
2
x
1
jrjx
1
x
0
j;
jx
3
x
2
jrjx
2
x
1
jr
2
jx
1
x
0
j;
jx
4
x
3
jrjx
3
x
2
jr
3
jx
1
x
0
j:
Therefore,weconjecturethat
jx
n
x
n1
jr
n1
jx
1
x
0
j if n1:
(1.2.7)
ThisistrivialfornD1.Ifitistrueforsomen,then(1.2.6)and(1.2.7)implythat
jx
nC1
x
n
jr.r
n1
jx
1
x
0
j/; so jx
nC1
x
n
jr
n
jx
1
x
0
j;
whichisproposition(1.2.7)withnreplacedbynC1. Hence, , (1.2.7)istrueforevery
positiveintegern,byTheorem1.2.1.
Themajoreffortinaninductionproof(afterP
1
,P
2
,...,P
n
,... havebeenformulated)
isusuallydirectedtowardshowingthatP
n
impliesP
nC1
.However,itisimportanttoverify
P
1
,sinceP
n
mayimplyP
nC1
evenifsomeorallofthepropositionsP
1
,P
2
,...,P
n
,...
arefalse.
Example1.2.4
LetP
n
bethepropositionthat2n1isdivisibleby2. IfP
n
istrue
thenP
nC1
isalso,since
2nC1D.2n1/C2:
However,wecannotconcludethatP
n
istrueforn1.Infact,P
n
isfalseforeveryn.
Thefollowingformulationoftheprincipleofmathematicalinductionpermitsustostart
inductionproofswithanarbitraryinteger,ratherthan1,asrequiredinTheorem1.2.1.
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14 Chapter1
TheRealNumbers
Theorem1.2.2
Letn
0
beanyinteger.positive;negative;orzero/:LetP
n
0
;P
n
0
C1
;
...;P
n
;... bepropositions;oneforeachintegernn
0
;suchthat
(a)
P
n
0
istrueI
(b)
foreachintegernn
0
;P
n
impliesP
nC1
:
ThenP
n
istrueforeveryintegernn
0
:
Proof
Form1,letQ
m
bethepropositiondefinedbyQ
m
DP
mCn
0
1
.ThenQ
1
D
P
n
0
istrueby
(a)
.Ifm1andQ
m
DP
mCn
0
1
istrue,thenQ
mC1
DP
mCn
0
istrueby
(b)
withnreplacedbymCn
0
1.Therefore,Q
m
istrueforallm1byTheorem1.2.1
withP replacedbyQandnreplacedbym. ThisisequivalenttothestatementthatP
n
is
trueforallnn
0
.
Example1.2.5
ConsiderthepropositionP
n
that
3nC16>0:
IfP
n
istrue,thensoisP
nC1
,since
3.nC1/C16D3nC3C16
D.3nC16/C3>0C3(bytheinductionassumption)
>0:
Thesmallestn
0
forwhichP
n
0
istrueisn
0
D 5. Hence,P
n
istrueforn   5,by
Theorem1.2.2.
Example1.2.6
LetP
n
bethepropositionthat
nŠ3
n
>0:
IfP
n
istrue,then
.nC1/Š3
nC1
DnŠ.nC1/3
nC1
>3
n
.nC1/3
nC1
(bytheinductionassumption)
D3
n
.n2/:
Therefore,P
n
impliesP
nC1
ifn>2. Bytrialanderror,n
0
D7isthesmallestinteger
suchthatP
n
0
istrue;hence,P
n
istrueforn7,byTheorem1.2.2.
Thenexttheoremisausefulconsequenceoftheprincipleofmathematicalinduction.
Theorem1.2.3
Letn
0
beanyinteger.positive;negative;orzero/:LetP
n
0
;P
n
0
C1
;...;
P
n
;... bepropositions;oneforeachintegernn
0
;suchthat
(a)
P
n
0
istrueI
(b)
fornn
0
;P
nC1
istrueifP
n
0
;P
n
0
C1
;...;P
n
arealltrue.
ThenP
n
istruefornn
0
:
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Section1.2
MathematicalInduction
15
Proof
Fornn
0
,letQ
n
bethepropositionthatP
n
0
,P
n
0
C1
,...,P
n
arealltrue.Then
Q
n
0
istrueby
(a)
.SinceQ
n
impliesP
nC1
by
(b)
,andQ
nC1
istrueifQ
n
andP
nC1
are
bothtrue,Theorem1.2.2impliesthatQ
n
istrueforallnn
0
.Therefore,P
n
istruefor
allnn
0
.
Example1.2.7
Anintegerp>1isaprimeifitcannotbefactoredaspDrswhere
randsareintegersand1<r,s<p.Thus,2,3,5,7,and11areprimes,and,although4,
6,8,9,and10arenot,theyareproductsofprimes:
4D22; 6D23; 8D222; 9D33; 10D25:
Theseobservationssuggestthateachintegern2isaprimeoraproductofprimes.Let
thispropositionbeP
n
. ThenP
2
istrue,butneitherTheorem 1.2.1norTheorem1.2.2
apply,sinceP
n
doesnotimplyP
nC1
inanyobviousway. (Forexample,itisnotevident
from24D2223that25isaproductofprimes.)However,Theorem1.2.3yieldsthe
statedresult,asfollows. Supposethatn2andP
2
,...,P
n
aretrue. EithernC1isa
primeor
nC1Drs;
(1.2.8)
whererandsareintegersand1<r,s<n,soP
r
andP
s
aretruebyassumption.Hence,r
andsareprimesorproductsofprimesand(1.2.8)impliesthatnC1isaproductofprimes.
WehavenowprovedP
nC1
(thatnC1isaprimeoraproductofprimes).Therefore,P
n
is
trueforalln2,byTheorem1.2.3.
1.2Exercises
ProvetheassertionsinExercises1.2.11.2.6byinduction.
1.
Thesumofthefirstnoddintegersisn
2
.
2.
1
2
C2
2
CCn
2
D
n.nC1/.2nC1/
6
:
3.
1
2
C3
2
CC.2n1/
2
D
n.4n
2
1/
3
:
4.
Ifa
1
,a
2
,...,a
n
arearbitraryrealnumbers,then
ja
1
Ca
2
CCa
n
jja
1
jCja
2
jCCja
n
j:
5.
Ifa
i
0,i1,then
.1Ca
1
/.1Ca
2
/.1Ca
n
/1Ca
1
Ca
2
CCa
n
:
6.
If0a
i
1,i1,then
.1a
1
/.1a
2
/.1a
n
/1a
1
a
2
a
n
:
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16 Chapter1
TheRealNumbers
7.
Supposethats
0
>0ands
n
D1e
s
n1
,n1.Showthat0<s
n
<1,n1.
8.
SupposethatR>0,x
0
>0,and
x
nC1
D
1
2
R
x
n
Cx
n
; n0:
Prove:Forn1,x
n
>x
nC1
>
p
Rand
x
n
p
R
1
2n
.x
0
p
R/
2
x
0
:
9.
Findandprovebyinductionanexplicitformulafora
n
ifa
1
D1and,forn 1,
(a)
a
nC1
D
a
n
.nC1/.2nC1/
(b)
a
nC1
D
3a
n
.2nC2/.2nC3/
(c)
a
nC1
D
2nC1
nC1
a
n
(d)
a
nC1
D
1C
1
n
n
a
n
10.
Leta
1
D 0anda
nC1
D.nC1/a
n
forn1,andletP
n
bethepropositionthat
a
n
DnŠ
(a)
ShowthatP
n
impliesP
nC1
.
(b)
IsthereanintegernforwhichP
n
istrue?
11.
LetP
n
bethepropositionthat
1C2CCnD
.nC2/.n1/
2
:
(a)
ShowthatP
n
impliesP
nC1
.
(b)
IsthereanintegernforwhichP
n
istrue?
12.
Forwhatintegersnis
1
>
8n
.2n/Š
Proveyouranswerbyinduction.
13.
Letabeaninteger2.
(a)
Showbyinductionthatifnisanonnegativeinteger,thennDaqCr,where
q(quotient)andr(remainder)areintegersand0r<a.
(b)
Showthattheresultof
(a)
istrueifnisanarbitraryinteger(notnecessarily
nonnegative).
(c)
ShowthatthereisonlyonewaytowriteagivenintegernintheformnD
aqCr,whereqandrareintegersand0r<a.
14.
Takethefollowingstatementasgiven:Ifpisaprimeandaandbareintegerssuch
thatpdividestheproductab,thenpdividesaorb.
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Section1.2
MathematicalInduction
17
(a)
Prove:Ifp,p
1
,...,p
k
arepositiveprimesandpdividestheproductp
1
p
k
,
thenpDp
i
forsomeiinf1;:::;kg.
(b)
Letnbeaninteger> 1. Showthattheprimefactorizationofnfoundin
Example1.2.7isuniqueinthefollowingsense:If
nDp
1
p
r
and nDq
1
q
2
q
s
;
wherep
1
,...,p
r
,q
1
,...,q
s
arepositiveprimes,thenrDsandfq
1
;:::;q
r
g
isapermutationoffp
1
;:::;p
r
g.
15.
Leta
1
Da
2
D5and
a
nC1
Da
n
C6a
n1
; n2:
Showbyinductionthata
n
D3n.2/ifn1.
16.
Leta
1
D2,a
2
D0,a
3
D14,and
a
nC1
D9a
n
23a
n1
C15a
n2
; n3:
Showbyinductionthata
n
D3
n1
5
n1
C2,n1.
17.
TheFibonaccinumbersfF
n
g
1
nD1
aredefinedbyF
1
DF
2
D1and
F
nC1
DF
n
CF
n1
; n2:
Provebyinductionthat
F
n
D
.1C
p
5/n.1
p
5/n
2n
p
5
; n1:
18.
Provebyinductionthat
Z
1
0
y
n
.1y/
r
dyD
.rC1/.rC2/.rCnC1/
ifnisanonnegativeintegerandr>1.
19.
Supposethatmandnareintegers,with0mn.Thebinomialcoefficient
n
m
!
isthecoefficientoft
m
intheexpansionof.1Ct/
n
;thatis,
.1Ct/
n
D
n
X
mD0
n
m
!
t
m
:
Fromthisdefinitionitfollowsimmediatelythat
n
0
!
D
n
n
!
D1; n0:
Forconveniencewedefine
n
1
!
D
n
nC1
!
D0; n0:
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18 Chapter1
TheRealNumbers
(a)
Showthat
nC1
m
!
D
n
m
!
C
n
m1
!
; 0mn;
andusethistoshowbyinductiononnthat
n
m
!
D
mŠ.nm/Š
; 0mn:
(b)
Showthat
Xn
mD0
.1/
m
n
m
!
D0 and
Xn
mD0
n
m
!
D2
n
:
(c)
Showthat
.xCy/
n
D
Xn
mD0
n
m
!
x
m
y
nm
:
(Thisisthebinomialtheorem.)
20.
Useinductiontofindannthantiderivativeoflogx,thenaturallogarithmofx.
21.
Letf
1
.x
1
/Dg
1
.x
1
/Dx
1
.Forn2,let
f
n
.x
1
;x
2
;:::;x
n
/Df
n1
.x
1
;x
2
;:::;x
n1
/C2
n2
x
n
C
jf
n1
.x
1
;x
2
;:::;x
n1
/2
n2
x
n
j
and
g
n
.x
1
;x
2
;:::;x
n
/Dg
n1
.x
1
;x
2
;:::;x
n1
/C2
n2
x
n
jg
n1
.x
1
;x
2
;:::;x
n1
/2
n2
x
n
j:
Findexplicitformulasforf
n
.x
1
;x
2
;:::;x
n
/andg
n
.x
1
;x
2
;:::;x
n
/.
22.
Provebyinductionthat
sinxCsin3xCCsin.2n1/xD
1cos2nx
2sinx
; n1:
H
INT
:Youwillneedtrigonometricidentitiesthatyoucanderivefromtheidentities
cos.AB/DcosAcosBCsinAsinB;
cos.ACB/DcosAcosBsinAsinB:
Takethesetwoidentitiesasgiven:
Section1.3
TheRealLine
19
23.
Supposethata
1
a
2
a
n
andb
1
b
2
b
n
.Letf`
1
;`
2
;:::`
n
gbea
permutationoff1;2;:::;ng,anddefine
Q.`
1
;`
2
;:::;`
n
/D
Xn
iD1
.a
i
b
`
i
/
2
:
Showthat
Q.`
1
;`
2
;:::;`
n
/Q.1;2;:::;n/:
1.3THEREALLINE
Oneofourobjectivesistodeveloprigorouslytheconceptsoflimit,continuity,differen-
tiability,andintegrability,whichyouhaveseenincalculus. Todothisrequiresabetter
understandingoftherealnumbersthanisprovidedincalculus.Thepurposeofthissection
istodevelopthisunderstanding.Sincetheutilityoftheconceptsintroducedherewillnot
becomeapparentuntilwearewellintothestudyoflimitsandcontinuity,youshouldre-
servejudgmentontheirvalueuntiltheyareapplied.Asthisoccurs,youshouldrereadthe
applicablepartsofthissection.Thisappliesespeciallytotheconceptofanopencovering
andtotheHeine–BorelandBolzano–Weierstrasstheorems,whichwillseemmysteriousat
first.
Weassumethatyouarefamiliarwiththegeometricinterpretationoftherealnumbersas
pointsonaline.Wewillnotprovethatthisinterpretationislegitimate,fortworeasons:(1)
theproofrequiresanexcursionintothefoundationsofEuclideangeometry,whichisnot
thepurposeofthisbook;(2)althoughwewillusegeometricterminologyandintuitionin
discussingthereals,wewillbaseallproofsonproperties
(A)
(I)
(Section1.1)andtheir
consequences,notongeometricarguments.
Henceforth,wewillusethetermsrealnumbersystemandreallinesynonymouslyand
denotebothbythesymbolR;also,wewilloftenrefertoarealnumberasapoint(onthe
realline).
SomeSetTheory
Inthissectionweareinterestedinsetsofpointsontherealline;however,wewillconsider
otherkindsofsetsinsubsequentsections. Thefollowingdefinitionappliestoarbitrary
sets,withtheunderstandingthatthemembersofallsetsunderconsiderationinanygiven
contextcomefromaspecificcollectionofelements,calledtheuniversalset.Inthissection
theuniversalsetistherealnumbers.
Definition1.3.1
LetSandT besets.
(a)
ScontainsT,andwewriteSTorT S,ifeverymemberofTisalsoinS.In
thiscase,T isasubsetofS.
(b)
STisthesetofelementsthatareinSbutnotinT.
(c)
SequalsT,andwewriteSDT,ifScontainsTandTcontainsS;thus,S DT if
andonlyifSandT havethesamemembers.
20 Chapter1
TheRealNumbers
(d)
S strictlycontainsT ifS S containsT butT doesnotcontainS; ; thatis, , ifevery
memberofTisalsoinS,butatleastonememberofSisnotinT (Figure1.3.1).
(e)
ThecomplementofS,denotedbyS
c
,isthesetofelementsintheuniversalsetthat
arenotinS.
(f)
TheunionofSandT,denotedbyS[T,isthesetofelementsinatleastoneofS
andT(Figure1.3.1
(b)
).
(g)
TheintersectionofSandT,denotedbyS \T,isthesetofelementsinbothSand
T (Figure1.3.1
(c)
). IfS\T D;(theemptyset),thenSandT T aredisjointsets
(Figure1.3.1
(d)
).
(h)
Asetwithonlyonememberx
0
isasingletonset,denotedbyfx
0
g.
T
S
   T
(a)
S ∪ T = shaded region
(b)
(c)
(d)
S ∩ T = shaded region
S ∩ T = ∅
T
S
T
S
T
S
Figure1.3.1
Example1.3.1
Let
SD
˚
x
ˇ
ˇ
0<x<1
; TD
˚
x
ˇ
ˇ
0<x<1andxisrational
;
and
U D
˚
x
ˇ
ˇ
0<x<1andxisirrational
:
ThenSTandSU,andtheinclusionisstrictinbothcases.Theunionsofpairsof
thesesetsare
S[T DS; S[U U DS; and T[U U DS;
andtheirintersectionsare
S\T DT; S\U U DU; and T\U U D;:
Section1.3
TheRealLine
21
Also,
SU DT
and ST T DU:
EverysetScontainstheemptyset;,fortosaythat;isnotcontainedinSistosaythat
somememberof;isnotinS,whichisabsurdsince;hasnomembers. IfSisanyset,
then
.S
c
/
c
DS and S\S
c
D;:
IfSisasetofrealnumbers,thenS[S
c
DR.
Thedefinitionsofunionandintersectionhavegeneralizations:IfF isanarbitrarycol-
lectionofsets,then[
˚
S
ˇ
ˇ
S2F
isthesetofallelementsthataremembersofatleast
oneofthesetsinF,and\
˚
S
ˇ
ˇ
S2F
isthesetofallelementsthataremembersofevery
setinF. TheunionandintersectionoffinitelymanysetsS
1
,...,S
n
arealsowrittenas
S
n
kD1
S
k
and
T
n
kD1
S
k
.TheunionandintersectionofaninfinitesequencefS
k
g
1
kD1
ofsets
arewrittenas
S
1
kD1
S
k
and
T
1
kD1
S
k
.
Example1.3.2
IfF isthecollectionofsets
S
D
˚
x
ˇ
ˇ
<x1C
; 0<1=2;
then
[
˚
S
ˇ
ˇ
S
2F
D
˚
x
ˇ
ˇ
0<x3=2
and
\
˚
S
ˇ
ˇ
S
2F
D
˚
x
ˇ
ˇ
1=2<x1
:
Example1.3.3
If,foreachpositiveintegerk,thesetS
k
isthesetofrealnumbers
thatcanbewrittenasx D m=kforsomeintegerm,then
S
1
kD1
S
k
isthesetofrational
numbersand
T
1
kD1
S
k
isthesetofintegers.
OpenandClosedSets
Ifaandbareintheextendedrealsanda<b,thentheopeninterval.a;b/isdefinedby
.a;b/D
˚
x
ˇ
ˇ
a<x<b
:
Theopenintervals.a;1/and.1;b/aresemi-infiniteifaandbarefinite,and.1;1/
istheentirerealline.
Definition1.3.2
Ifx
0
isarealnumberand>0,thentheopeninterval.x
0
;x
0
C/
isan-neighborhoodofx
0
. IfasetS S containsan-neighborhoodofx
0
, thenS isa
neighborhoodofx
0
,andx
0
isaninteriorpointofS (Figure1.3.2). Thesetofinterior
pointsofSistheinteriorofS,denotedbyS
0
.IfeverypointofSisaninteriorpoint(that
is,S
0
DS),thenSisopen.AsetSisclosedifS
c
isopen.
22 Chapter1
TheRealNumbers
(
)
x
0
x
0
− 
x
0
x
0
= interior point of S 
S = four line segments
Figure1.3.2
Theideaofneighborhoodisfundamentalandoccursinmanyothercontexts,someof
whichwewillseelaterinthisbook.Whateverthecontext,theideaisthesame:somedefi-
nitionof“closeness”isgiven(forexample,tworealnumbersare“close”iftheirdifference
is“small”),andaneighborhoodofapointx
0
isasetthatcontainsallpointssufficiently
closetox
0
.
Example1.3.4
Anopeninterval.a;b/is anopenset, because ifx
0
2 .a;b/and
minfx
0
a;bx
0
g,then
.x
0
;x
0
C/.a;b/:
TheentirelineRD.1;1/isopen,andtherefore;.D Rc/isclosed. However,;is
alsoopen,fortodenythisistosaythat;containsapointthatisnotaninteriorpoint,
whichisabsurdbecause;containsnopoints.Since;isopen,R.D;
c
/isclosed.Thus,
Rand;arebothopenandclosed. TheyaretheonlysubsetsofRwiththisproperty
(Exercise1.3.18).
Adeletedneighborhoodofapointx
0
isasetthatcontainseverypointofsomeneigh-
borhoodofx
0
exceptforx
0
itself.Forexample,
S D
˚
x
ˇ
ˇ
0<jxx
0
j<
isadeletedneighborhoodofx
0
.Wealsosaythatitisadeleted-neighborhoodofx
0
.
Theorem1.3.3
(a)
Theunionofopensetsisopen:
(b)
Theintersectionofclosedsetsisclosed:
Thesestatementsapplytoarbitrarycollections,finiteorinfinite,ofopenandclosedsets:
Proof (a)
LetGbeacollectionofopensetsand
SD[
˚
G
ˇ
ˇ
G2G
:
Ifx
0
2 S,thenx
0
2 G
0
forsomeG
0
inG,andsinceG
0
isopen,itcontainssome-
neighborhoodofx
0
.SinceG
0
S,this-neighborhoodisinS,whichisconsequentlya
neighborhoodofx
0
. Thus,Sisaneighborhoodofeachofitspoints,andthereforeopen,
bydefinition.
(b)
Let F bea a collectionofclosed setsandT D \
˚
F
ˇ
ˇ
F 2F
. Then n T
c
D
[
˚
F
c
ˇ
ˇ
F2F
(Exercise1.3.7)and,sinceeachF
c
isopen,T
c
isopen,from
(a)
.There-
fore,T isclosed,bydefinition.
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