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Section3.5
AdvancedLookattheExistenceoftheProperRiemannIntegral
173
(Exercise3.5.1).Ifa<x<b,theoscillationoff atxisdefinedby
w
f
.x/D lim
h!0C
W
f
.xh;xCh/:
ThecorrespondingdefinitionsforxDaandxDbare
w
f
.a/D lim
h!0C
W
f
.a;aCh/ and
w
f
.b/D lim
h!0C
W
f
.bh;b/:
Forafixedxin.a;b/,W
f
.xh;xCh/isanonnegativeandnondecreasingfunction
ofhfor0 < < h < min.xa;bx/; ; therefore,w
f
.x/existsandisnonnegative, by
Theorem2.1.9.Similarargumentsapplytow
f
.a/andw
f
.b/.
Theorem3.5.2
Letf bedefinedonŒa;b:Thenf iscontinuousatx
0
inŒa;bif
andonlyifw
f
.x
0
/ D D 0:.Continuityataorbmeanscontinuityfromtherightorleft,
respectively./
Proof
Supposethata<x
0
<b.First,supposethatw
f
.x
0
/D0and>0.Then
W
f
Œx
0
h;x
0
Ch<
forsomeh>0,so
jf.x/f.x
0
/j< if x
0
hx;x
0
x
0
Ch:
Lettingx
0
Dx
0
,weconcludethat
jf.x/f.x
0
/j< if jxx
0
j<h:
Therefore,f iscontinuousatx
0
.
Conversely,iff iscontinuousatx
0
and>0,thereisaı>0suchthat
jf.x/f.x
0
/j<
2
and jf.x
0
/f.x
0
/j<
2
ifx
0
ıx,xx
0
Cı.Fromthetriangleinequality,
jf.x/f.x
0
/jjf.x/f.x
0
/jCjf.x
0
/f.x
0
/j<;
so
W
f
Œx
0
h;x
0
Ch if h<ıI
therefore,w
f
.x
0
/D0.Similarargumentsapplyifx
0
Daorx
0
Db.
Lemma3.5.3
If w
f
.x/ <   for a  x  b; ; thenthere is aı > 0suchthat
W
f
Œa
1
;b
1
;providedthatŒa
1
;b
1
Œa;bandb
1
a
1
<ı:
Proof
WeusetheHeine–Boreltheorem(Theorem1.3.7). Ifw
f
.x/ < ,thereisan
h
x
>0suchthat
jf.x
0
/f.x
00
/j<
(3.5.1)
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174 Chapter3
IntegralCalculusofFunctionsofOneVariable
if
x2h
x
<x
0
;x
00
<xC2h
x
and x
0
;x
00
2Œa;b:
(3.5.2)
IfI
x
D.xh
x
;xCh
x
/,thenthecollection
HD
˚
I
x
ˇ
ˇ
axb
isanopencoveringofŒa;b,sotheHeine–Boreltheoremimpliesthattherearefinitely
manypointsx
1
,x
2
,...,x
n
inŒa;bsuchthatI
x
1
,I
x
2
,...,I
x
n
coverŒa;b.Let
hD min
1in
h
x
i
andsupposethatŒa
1
;b
1
  Œa;bandb
1
a
1
< h. Ifx
0
andx
00
areinŒa
1
;b
1
,then
x
0
2I
x
r
forsomer.1rn/,so
jx
0
x
r
j<h
x
r
:
Therefore,
jx
00
x
r
jjx
00
x
0
jCjx
0
x
r
j<b
1
a
1
Ch
x
r
< hCh
x
r
2h
x
r
:
Thus,anytwopointsxandx00 inŒa
1
;b
1
satisfy(3.5.2)withxDx
r
,sotheyalsosatisfy
(3.5.1).Therefore,isanupperboundfortheset
˚
jf.x
0
/f.x
00
/j
ˇ
ˇ
x
0
;x
00
2Œa
1
;b
1
;
whichhasthesupremumW
f
Œa
1
;b
1
.Hence,W
f
Œa
1
;b
1
.
Inthefollowing,L.I/isthelengthoftheintervalI.
Lemma3.5.4
Letf beboundedonŒa;banddefine
E
D
˚
x2Œa;b
ˇ
ˇ
w
f
.x/
:
ThenE
isclosed;andf isintegrableonŒa;bifandonlyifforeverypairofpositive
numbersandı;E
canbecoveredbyfinitelymanyopenintervalsI
1
;I
2
;...;I
p
such
that
Xp
jD1
L.I
j
/<ı:
(3.5.3)
Proof
WefirstshowthatE
isclosed.Supposethatx
0
isalimitpointofE
.Ifh>0,
thereisan
xfromE
in.x
0
h;x
0
Ch/. SinceŒ
xh
1
;
xCh
1
Œx
0
h;x
0
Chfor
sufficientlysmallh
1
andW
f
Œ
xh
1
;
xCh
1
,itfollowsthatW
f
Œx
0
h;x
0
Ch
forallh>0.Thisimpliesthatx
0
2E
,soE
isclosed(Corollary1.3.6).
Nowwewillshowthatthestatedconditioninnecessaryforintegrability.Supposethat
theconditionisnotsatisfied;thatis,thereisa>0andaı>0suchthat
p
X
jD1
L.I
j
/ı
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Section3.5
AdvancedLookattheExistenceoftheProperRiemannIntegral
175
foreveryfinitesetfI
1
;I
2
;:::;I
p
gofopenintervalscoveringE
.IfP Dfx
0
;x
1
;:::;x
n
g
isapartitionofŒa;b,then
S.P/s.P/D
X
j2A
.M
j
m
j
/.x
j
x
j1
/C
X
j2B
.M
j
m
j
/.x
j
x
j1
/; (3.5.4)
where
AD
˚
j
ˇ
ˇ
Œx
j1
;x
j
\E
¤;
and BD
˚
j
ˇ
ˇ
Œx
j1
;x
j
\E
D;
:
Since
S
j2A
.x
j1
;x
j
/containsallpointsofE
exceptanyofx
0
,x
1
,...,x
n
thatmay
beinE
,andeachofthesefinitelymanypossibleexceptionscanbecoveredbyanopen
intervaloflengthassmallasweplease,ourassumptiononE
impliesthat
X
j2A
.x
j
x
j1
/ı:
Moreover,ifj 2A,then
M
j
m
j
;
so(3.5.4)impliesthat
S.P/s.P/
X
j2A
.x
j
x
j1
/ı:
SincethisholdsforeverypartitionofŒa;b,fisnotintegrableonŒa;b,byTheorem3.2.7.
Thisprovesthatthestatedconditionisnecessaryforintegrability.
Forsufficiency,letandıbepositivenumbersandletI
1
,I
2
,...,I
p
beopenintervals
thatcoverE
andsatisfy(3.5.3).Let
e
I
j
DŒa;b\
I
j
:
(
I
j
DclosureofI.)Aftercombininganyof
e
I
1
,
e
I
2
,...,
e
I
p
thatoverlap,weobtainaset
ofpairwisedisjointclosedsubintervals
C
j
DŒ˛
j
j
; 1j j q.p/;
ofŒa;bsuchthat
a˛
1
1
2
2
<˛
q1
q1
q
q
b;
(3.5.5)
q
X
iD1
i
˛
i
/<ı
(3.5.6)
and
w
f
.x/<; ˇ
j
x˛
jC1
; 1j j q1:
Also,w
f
.x/<forax˛
1
ifa<˛
1
andforˇ
q
xbifˇ
q
<b.
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176 Chapter3
IntegralCalculusofFunctionsofOneVariable
LetP
0
bethepartitionofŒa;bwiththepartitionpointsindicatedin(3.5.5),andrefine
P
0
bypartitioningeachsubintervalŒˇ
j
jC1
(aswellasŒa;˛
1
ifa < < ˛
1
andŒˇ
q
;b
ifˇ
q
<b)intosubintervalsonwhichtheoscillationoff isnotgreaterthan. . Thisis
possiblebyLemma3.5.3. Inthisway, , afterrenamingtheentirecollectionofpartition
points,weobtainapartitionPDfx
0
;x
1
;:::;x
n
gofŒa;bforwhichS.P/s.P/canbe
writtenasin(3.5.4),with
X
j2A
.x
j
x
j1
/D
q
X
iD1
i
˛
i
/<ı
(see(3.5.6))and
M
j
m
j
; j j 2B:
Forthispartition,
X
j2A
.M
j
m
j
/.x
j
x
j1
/2K
X
j2A
.x
j
x
j1
/<2Kı;
whereKisanupperboundforjfjonŒa;band
X
j2B
.M
j
m
j
/.x
j
x
j1
/.ba/:
Wehavenowshownthatifandıarearbitrarypositivenumbers,thereisapartitionP of
Œa;bsuchthat
S.P/s.P/<2KıC.ba/:
(3.5.7)
If>0,let
ıD
4K
and D
2.ba/
:
Then(3.5.7)yields
S.P/s.P/<;
andTheorem3.2.7impliesthatf isintegrableonŒa;b.
WeneedthenextdefinitiontostateLebesgue’sintegrabilitycondition.
Definition3.5.5
AsubsetS ofthereallineisofLebesguemeasurezeroifforevery
>0thereisafiniteorinfinitesequenceofopenintervalsI
1
,I
2
,...suchthat
S
[
j
I
j
(3.5.8)
and
Xn
jD1
L.I
j
/<; n1:
(3.5.9)
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Section3.5
AdvancedLookattheExistenceoftheProperRiemannIntegral
177
NotethatanysubsetofasetofLebesguemeasurezeroisalsoofLebesguemeasurezero.
(Why?)
Example3.5.1
TheemptysetisofLebesguemeasurezero,sinceitiscontainedin
anyopeninterval.
Example3.5.2
AnyfinitesetS D D fx
1
;x
2
;:::;x
n
g isofLebesguemeasure zero,
sincewecanchooseopenintervalsI
1
,I
2
,...,I
n
suchthatx
j
2 I
j
andL.I
j
/< =n,
1j n.
Example3.5.3
Aninfinitesetisdenumerableifitsmemberscanbelistedinase-
quence(thatis,inaone-to-onecorrespondencewiththepositiveintegers);thus,
SDfx
1
;x
2
;:::;x
n
;:::g:
(3.5.10)
Aninfinitesetthatdoesnothavethispropertyisnondenumerable. Anydenumerableset
(3.5.10)isofLebesguemeasurezero,sinceif>0,itispossibletochooseopenintervals
I
1
,I
2
,...,sothatx
j
2I
j
andL.I
j
/<2j,j 1.Then(3.5.9)holdsbecause
1
2
C
1
22
C
1
23
CC
1
2n
D1
1
2n
<1:
(3.5.11)
TherearealsonondenumerablesetsofLebesguemeasurezero,butitisbeyondthescope
ofthisbooktodiscussexamples.
Thenexttheoremisthemainresultofthissection.
Theorem3.5.6
Aboundedfunctionf isintegrableonafiniteintervalŒa;bifand
onlyifthesetSofdiscontinuitiesoff inŒa;bisofLebesguemeasurezero:
Proof
FromTheorem3.5.2,
S D
˚
x2Œa;b
ˇ
ˇ
w
f
.x/>0
:
Sincew
f
.x/>0ifandonlyifw
f
.x/1=iforsomepositiveintegeri,wecanwrite
SD
[1
iD1
S
i
;
(3.5.12)
where
S
i
D
˚
x2Œa;b
ˇ
ˇ
w
f
.x/1=i
:
Nowsupposethatf isintegrableonŒa;band>0.FromLemma3.5.4,eachS
i
can
becoveredbyafinitenumberofopenintervalsI
i1
,I
i2
,...,I
in
oftotallengthlessthan
=2i.Wesimplyrenumbertheseintervalsconsecutively;thus,
I
1
;I
2
;DI
11
;:::;I
1n
1
;I
21
;:::;I
2n
2
;:::;I
i1
;:::;I
in
i
;::::
Now(3.5.8)and(3.5.9)holdbecauseof(3.5.11)and(3.5.12),andwehaveshownthatthe
statedconditionisnecessaryforintegrability.
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178 Chapter3
IntegralCalculusofFunctionsofOneVariable
Forsufficiency, supposethatthestatedconditionholdsand > 0. ThenS can n be
coveredbyopenintervalsI
1
;I
2
;:::thatsatisfy(3.5.9).If>0,thentheset
E
D
˚
x2Œa;b
ˇ
ˇ
w
f
.x/
ofLemma3.5.4iscontainedinS(Theorem3.5.2),andthereforeE
iscoveredbyI
1
;I
2
;:::.
SinceE
isclosed(Lemma3.5.4)andbounded,theHeine–BoreltheoremimpliesthatE
iscoveredbyafinitenumberofintervalsfromI
1
;I
2
;:::. Thesumofthelengthsofthe
latterislessthan,soLemma3.5.4impliesthatf isintegrableonŒa;b.
3.5Exercises
1.
InconnectionwithDefinition3.5.1,showthat
sup
x;x
0
2Œa;b
jf.x/f.x
0
/jD sup
axb
f.x/ inf
axb
f.x/:
2.
UseTheorem3.5.6toshowthatiff isintegrableonŒa;b,thensoisjfjand,if
f.x/>0.axb/,sois1=f.
3.
Prove: TheunionoftwosetsofLebesguemeasurezeroisofLebesguemeasure
zero.
4.
UseTheorem3.5.6andExercise3.5.3toshowthatiff andgareintegrableon
Œa;b,thensoarefCgandfg.
5.
Supposef isintegrableonŒa;b,˛D D inf
axb
f.x/,andˇ D sup
axb
f.x/.
LetgbecontinuousonŒ˛;ˇ.ShowthatthecompositionhDgıf isintegrable
onŒa;b.
6.
Letf beintegrableonŒa;b,let˛Dinf
axb
f.x/andˇDsup
axb
f.x/,and
supposethatGiscontinuousonŒ˛;ˇ.Foreachn1,let
aC
.j 1/.ba/
n
u
jn
;v
jn
aC
j.ba/
n
; 1j j n:
Showthat
lim
n!1
1
n
Xn
jD1
jG.f.u
jn
//G.f.v
jn
//jD0:
7.
Leth.x/ D 0forallx inŒa;bexceptforx inasetofLebesguemeasurezero.
Showthatif
R
b
a
h.x/dxexists,itequalszero.H
INT
:Anysubsetofasetofmeasure
zeroisalsoofmeasurezero:
8.
Supposethatf andgareintegrableonŒa;bandf.x/Dg.x/exceptforxinaset
ofLebesguemeasurezero.Showthat
Z
b
a
f.x/dxD
Z
b
a
g.x/dx:
CHAPTER4
InfiniteSequences andSeries
INTHISCHAPTERweconsiderinfinitesequencesandseriesofconstantsandfunctions
ofarealvariable.
SECTION4.1introducesinfinitesequencesofrealnumbers. Theconceptofalimitofa
sequenceisdefined,asistheconceptofdivergenceofasequenceto˙1. Wediscuss
boundedsequencesandmonotonicsequences. Thelimitinferiorandlimitsuperiorofa
sequencearedefined. WeprovetheCauchyconvergencecriterionforsequencesofreal
numbers.
SECTION4.2definesasubsequenceofaninfinitesequence. Weshowthatifasequence
convergestoalimitordivergesto˙1,thensodoallsubsequencesofthesequence.Limit
pointsandboundednessofasetofrealnumbersarediscussedintermsofsequencesof
membersoftheset.Continuityandboundednessofafunctionarediscussedintermsofthe
valuesofthefunctionatsequencesofpointsinitsdomain.
SECTION4.3introducesconceptsofconvergenceanddivergenceto˙1forinfiniteseries
ofconstants. WeproveCauchy’sconvergencecriterionforaseriesofconstants. . Incon-
nectionwithseriesofpositiveterms,weconsiderthecomparisontest,theintegraltest,the
ratiotest,andRaabe’stest. Forgeneralseries,weconsiderabsoluteandconditionalcon-
vergence,Dirichlet’stest,rearrangementofterms,andmultiplicationofoneinfiniteseries
byanother.
SECTION4.4dealswithpointwiseanduniformconvergenceofsequencesandseriesof
functions. Cauchy’suniformconvergencecriteriaforsequencesandseriesareproved,as
isDirichlet’stestforuniformconvergenceofaseries. Wegivesufficientconditionsfor
thelimitofasequenceoffunctionsorthesum ofaninfiniteseriesoffunctionstobe
continuous,integrable,ordifferentiable.
SECTION4.5considerspowerseries. Itisshownthatapowerseriesthatconvergeson
anopenintervaldefinesaninfinitelydifferentiablefunctiononthatinterval. Wedefine
theTaylorseriesofaninfinitelydifferentiablefunction,andgivesufficientconditionsfor
theTaylorseriestoconvergetothefunctiononsomeinterval.Arithmeticoperationswith
powerseriesarediscussed.
178
Section4.1
SequencesofRealNumbers
179
4.1SEQUENCESOFREALNUMBERS
Aninfinitesequence(morebriefly,asequence)ofrealnumbersisareal-valuedfunction
definedonasetofintegers
˚
n
ˇ
ˇ
nk
.Wecallthevaluesofthefunctionthetermsofthe
sequence.Wedenoteasequencebylistingitstermsinorder;thus,
fs
n
g
1
k
Dfs
k
;s
kC1
;:::g:
(4.1.1)
Forexample,
1
n2C1
1
0
D
1;
1
2
;
1
5
;:::;
1
n2C1
;:::
;
f.1/
n
g
1
0
Df1;1;1;:::;.1/
n
;:::g;
and
1
n2
1
3
D
1;
1
2
;
1
3
;:::;
1
n2
;:::
:
Therealnumbers
n
isthenthtermofthesequence. Usuallyweareinterestedonlyinthe
termsofasequenceandtheorderinwhichtheyappear,butnotintheparticularvalueofk
in(4.1.1).Therefore,weregardthesequences
1
n2
1
3
and
1
n
1
1
asidentical.
Wewillusuallywritefs
n
gratherthanfs
n
g
1
k
. Intheabsenceofanyindicationtothe
contrary,wetakekD0unlesss
n
isgivenbyarulethatisinvalidforsomenonnegative
integer,inwhichcasekisunderstoodtobethesmallestpositiveintegersuchthats
n
is
definedforallnk.Forexample,if
s
n
D
1
.n1/.n5/
;
thenkD6.
Theinterestingquestionsaboutasequencefs
n
gconcernthebehaviorofs
n
forlargen.
LimitofaSequence
Definition4.1.1
Asequencefs
n
gconvergestoalimitsifforevery>0thereisan
integerNsuchthat
js
n
sj< if nN:
(4.1.2)
Inthiscasewesaythatfs
n
gisconvergentandwrite
lim
n!1
s
n
Ds:
Asequencethatdoesnotconvergediverges,orisdivergent
180 Chapter4
InfiniteSequencesandSeries
AswesawinSection2.1whendiscussinglimitsoffunctions,Definition4.1.1isnot
changedbyreplacing(4.1.2)with
js
n
sj<K if nN;
whereKisapositiveconstant.
Example4.1.1
Ifs
n
Dcfornk,thenjs
n
cjD0fornk,andlim
n!1
s
n
Dc.
Example4.1.2
If
s
n
D
2nC1
nC1
;
thenlim
n!1
s
n
D2,since
js
n
2jD
ˇ
ˇ
ˇ
ˇ
2nC1
nC1
2nC2
nC1
ˇ
ˇ
ˇ
ˇ
D
1
nC1
I
hence,if>0,then(4.1.2)holdswithsD2ifN 1=.
Definition4.1.1doesnotrequirethattherebeanintegerN suchthat(4.1.2)holdsfor
all;rather,itrequiresthatforeachpositivetherebeanintegerN thatsatisfies(4.1.2)
forthatparticular.Usually,Ndependsonandmustbeincreasedifisdecreased.The
constantsequences (Example4.1.1)areessentiallytheonlyonesforwhichN N doesnot
dependon(Exercise4.1.5).
Wesaythatthetermsofasequencefs
n
g
1
k
satisfyagivenconditionforallnifs
n
satisfies
theconditionforallnk,orforlargenifthereisanintegerN >ksuchthats
n
satisfies
theconditionwhenevernN. Forexample,thetermsoff1=ng
1
1
arepositiveforalln,
whilethoseoff17=ng
1
1
arepositiveforlargen(takeND8).
Uniqueness oftheLimit
Theorem4.1.2
Thelimitofaconvergentsequenceisunique:
Proof
Supposethat
lim
n!1
s
n
Ds and
lim
n!1
s
n
Ds
0
:
WemustshowthatsDs
0
.Let>0.FromDefinition4.1.1,thereareintegersN
1
andN
2
suchthat
js
n
sj< if nN
1
(becauselim
n!1
s
n
Ds),and
js
n
s
0
j< if nN
2
Section4.1
SequencesofRealNumbers
181
(becauselim
n!1
s
n
Ds
0
).TheseinequalitiesbothholdifnN Dmax.N
1
;N
2
/,which
impliesthat
jss
0
jDj.ss
N
/C.s
N
s
0
/j
jss
N
jCjs
N
s
0
j<CD2:
Sincethisinequalityholdsforevery>0andjss
0
jisindependentof,weconclude
thatjss
0
jD0;thatis,sDs
0
.
Sequences Divergingto
˙1
Wesaythat
lim
n!1
s
n
D1
ifforanyrealnumbera,s
n
>aforlargen.Similarly,
lim
n!1
s
n
D1
ifforanyrealnumbera,s
n
<aforlargen.However,wedonotregardfs
n
gasconvergent
unlesslim
n!1
s
n
isfinite,asrequiredbyDefinition4.1.1. Toemphasizethisdistinction,
wesaythatfs
n
gdivergesto1.1/iflim
n!1
s
n
D1.1/.
Example4.1.3
Thesequencefn=2C1=ngdivergesto1,since,ifaisanyrealnum-
ber,then
n
2
C
1
n
>a if n2a:
Thesequencefnn
2
gdivergesto1,since,ifaisanyrealnumber,then
n
2
CnDn.n1/<a if n>1C
p
jaj:
Therefore,wewrite
lim
n!1
n
2
C
1
n
D1
and
lim
n!1
.n
2
Cn/D1:
Thesequencef.1/
n
n
3
gdiverges,butnotto1or1.
BoundedSequences
Definition4.1.3
Asequencefs
n
gisboundedaboveifthereisarealnumberbsuch
that
s
n
b foralln;
boundedbelowifthereisarealnumberasuchthat
s
n
a foralln;
orboundedifthereisarealnumberrsuchthat
js
n
jr foralln:
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