292 Chapter5
Real-ValuedFunctionsof
n
Variables
TheHeine–Boreltheorem(Theorem1.3.7)alsoholdsinR
n
,buttheproofinSection1.3
isvalidonlyfornD1. ToprovetheHeine–Boreltheoremforgeneraln,weneedsome
preliminarydefinitionsandresultsthatareofinterestintheirownright.
Definition5.1.13
AsequenceofpointsfX
r
ginR
n
convergestothelimit
Xif
lim
r!1
jX
r
XjD0:
Inthiscasewewrite
lim
r!1
X
r
D
X:
Thenexttwotheoremsfollowfromthis,thedefinitionofdistanceinR
n
,andwhatwe
alreadyknowaboutconvergenceinR. Weleavetheproofstoyou(Exercises5.1.16and
5.1.17).
Theorem5.1.14
Let
XD.
x
1
;
x
2
;:::;
x
n
/ and X
r
D.x
1r
;x
2r
;:::;x
nr
/; r1:
Thenlim
r!1
X
r
D
Xifandonlyif
lim
r!1
x
ir
D
x
i
; 1inI
thatis;asequencefX
r
gofpointsinRconvergestoalimit
Xifandonlyifthesequences
ofcomponentsoffX
r
gconvergetotherespectivecomponentsof
X:
Theorem5.1.15(Cauchy’sConvergenceCriterion)
AsequencefX
r
gin
R
n
convergesifandonlyifforeach>0thereisanintegerKsuchthat
jX
r
X
s
j< if r;sK:
Thenextdefinitiongeneralizesthedefinitionofthediameterofacircleorsphere.
Definition5.1.16
IfSisanonemptysubsetofR
n
,then
d.S/Dsup
˚
jXYj
ˇ
ˇ
X;Y2S
isthediameterofS.Ifd.S/<1;SisboundedIifd.S/D1,Sisunbounded.
Theorem5.1.17(PrincipleofNestedSets)
IfS
1
;S
2
;...areclosednonempty
subsetsofR
n
suchthat
S
1
S
2
S
r

(5.1.14)
and
lim
r!1
d.S
r
/D0;
(5.1.15)
thentheintersection
I D
\1
rD1
S
r
containsexactlyonepoint:
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Section5.1
Structureof
R
n
293
Proof
LetfX
r
gbeasequencesuchthatX
r
2S
r
.r1/.Becauseof(5.1.14),X
r
2S
k
ifrk,so
jX
r
X
s
j<d.S
k
/ if r;sk:
From(5.1.15)andTheorem5.1.15,X
r
convergestoalimit
X.Since
Xisalimitpointof
everyS
k
andeveryS
k
isclosed,
XisineveryS
k
(Corollary1.3.6).Therefore,
X2I,so
I ¤;.Moreover,
XistheonlypointinI,sinceifY2I,then
j
XYjd.S
k
/; k1;
and(5.1.15)impliesthatYD
X.
WecannowprovetheHeine–BoreltheoremforR
n
. Thistheoremconcernscompact
sets.AsinR,acompactsetinRisaclosedandboundedset.
RecallthatacollectionHofopensetsisanopencoveringofasetSif
S[
˚
H
ˇ
ˇ
H 2H
:
Theorem5.1.18(Heine–Borel Theorem)
IfHisanopencoveringofacom-
pactsubsetS;thenScanbecoveredbyfinitelymanysetsfromH:
Proof
Theproofisbycontradiction. WefirstconsiderthecasewherenD2,sothat
youcanvisualizethemethod. SupposethatthereisacoveringHforS fromwhichitis
impossibletoselectafinitesubcovering. SinceSisbounded,Siscontainedinaclosed
square
T Df.x;y/ja
1
xa
1
CL;a
2
xa
2
CLg
withsidesoflengthL(Figure5.1.6).
T(1)
S(1)
S(2)
S(3)
S(4)
T(2)
T(3)
T(4)
Figure5.1.6
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294 Chapter5
Real-ValuedFunctionsof
n
Variables
BisectingthesidesofTasshownbythedashedlinesinFigure5.1.6leadstofourclosed
squares,T
.1/
;T
.2/
,T
.3/
,andT
.4/
,withsidesoflengthL=2.Let
S
.i/
DS\T
.i/
; 1i4:
EachS
.i/
,beingtheintersectionofclosedsets,isclosed,and
SD
[4
iD1
S
.i/
:
Moreover,HcoverseachS
.i/
,butatleastoneS
.i/
cannotbecoveredbyanyfinitesub-
collectionofH,sinceifalltheS
.i/
couldbe,thensocouldS. LetS
1
beasetwiththis
property,chosenfromS
.1/
,S
.2/
, S
.3/
,andS
.4/
. Wearenowbacktothesituationwe
startedfrom: acompactsetS
1
coveredbyH,butnotbyanyfinitesubcollectionofH.
However,S
1
iscontainedinasquareT
1
withsidesoflengthL=2insteadofL. Bisecting
thesidesofT
1
andrepeatingtheargument,weobtainasubsetS
2
ofS
1
thathasthesame
propertiesasS,exceptthatitiscontainedinasquarewithsidesoflengthL=4.Continuing
inthiswayproducesasequenceofnonemptyclosedsetsS
0
.DS/,S
1
,S
2
,...,suchthat
S
k
S
kC1
andd.S
k
/L=2
k1=2
.k0/. FromTheorem5.1.17,thereisapoint
Xin
T
1
kD1
S
k
.Since
X2S,thereisanopensetHinHthatcontains
X,andthisHmustalso
containsome-neighborhoodof
X.SinceeveryXinS
k
satisfiestheinequality
jX
Xj2
kC1=2
L;
itfollowsthatS
k
 H fork sufficientlylarge. ThiscontradictsourassumptiononH,
whichledustobelievethatnoS
k
couldbecoveredbyafinitenumberofsetsfromH.
Consequently,thisassumptionmustbefalse:Hmusthaveafinitesubcollectionthatcovers
S.ThiscompletestheprooffornD2.
Theideaoftheproofisthesameforn> 2. ThecounterpartofthesquareT T isthe
hypercubewithsidesoflengthL:
T D
˚
.x
1
;x
2
;:::;x
n
/
ˇ
ˇ
a
i
x
i
a
i
CL;iD1;2;:::;n
:
Halvingtheintervalsofvariationofthencoordinatesx
1
, x
2
,..., x
n
dividesT into2
n
closedhypercubeswithsidesoflengthL=2:
T
.i/
D
˚
.x
1
;x
2
;:::;x
n
/
ˇ
ˇ
b
i
x
i
b
i
CL=2;1in
;
whereb
i
Da
i
orb
i
Da
i
CL=2. IfnofinitesubcollectionofHcoversS,thenatleast
oneofthesesmallerhypercubesmustcontainasubsetofSthatisnotcoveredbyanyfinite
subcollectionofS.NowtheproofproceedsasfornD2.
TheBolzano–WeierstrasstheoremisvalidinR
n
;itsproofisthesameasinR.
Connected SetsandRegions
Althoughitislegitimatetoconsiderfunctionsdefinedonarbitrarydomains,werestricted
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Section5.1
Structureof
R
n
295
ourstudyoffunctionsofonevariablemainlytofunctionsdefinedonintervals. Thereare
goodreasonsforthis. Ifwewishtoraisequestionsofcontinuityanddifferentiabilityat
everypointofthedomainDofafunctionf,theneverypointofDmustbealimitpoint
ofD
0
. Intervalshavethisproperty. Moreover,thedefinitionof
R
b
a
f.x/dxisobviously
applicableonlyiff isdefinedonŒa;b.
Itisnotproductivetoconsiderquestionsofcontinuityanddifferentiabilityoffunctions
definedontheunionofdisjointintervals,sincemanyimportantresultssimplydonothold
forsuchdomains.Forexample,theintermediatevaluetheorem(Theorem2.2.10;seealso
Exercise2.2.25)saysthatiff iscontinuousonanintervalI I andf.x
1
/ < < f.x
2
/
forsomex
1
andx
2
inI,thenf.
x/Dforsome
xinI. Theorem2.3.12saysthatf f is
constantonanintervalIiff0onI.NeitheroftheseresultsholdsifI istheunionof
disjointintervalsratherthanasingleinterval;thus,iff isdefinedonI I D.0;1/[.2;3/
by
f.x/D
1; 0<x<1;
0; 2<x<3;
thenf iscontinuousonI,butdoesnotassumeanyvaluebetween0and1,andf0on
I,butf isnotconstant.
Itisnotdifficulttoseewhytheseresultsfailtoholdforthisfunction:thedomainoff
consistsoftwodisconnectedpieces.Itwouldbemoresensibletoregardf astwoentirely
differentfunctions,onedefinedon.0;1/andtheotheron.2;3/.Thetworesultsmentioned
arevalidforeachofthesefunctions.
AswewillseewhenwestudyfunctionsdefinedonsubsetsofR
n
,considerationslike
thosejustcitedasmakingitnaturaltoconsiderfunctionsdefinedonintervalsinRlead
ustosingleoutapreferredclassofsubsetsasdomainsoffunctionsofnvariables.These
subsetsarecalledregions.Todefinethisterm,wefirstneedthefollowingdefinition.
Definition5.1.19
AsubsetSofR
n
isconnectedifitisimpossibletorepresentSas
theunionoftwodisjointnonemptysetssuchthatneithercontainsalimitpointoftheother;
thatis,ifScannotbeexpressedasSDA[B,where
A¤;; B¤;;
A\BD;; and A\
BD;:
(5.1.16)
IfScanbeexpressedinthisway,thenSisdisconnected.
Example5.1.8
Theemptysetandsingletonsetsareconnected,becausetheycannot
berepresentedastheunionoftwodisjointnonemptysets.
Example5.1.9
ThespaceR
n
isconnected,becauseifR
n
DA[Bwith
A\BD;
andA\
BD;,then
AAand
BB;thatis,AandBarebothclosedandtherefore
arebothopen. SincetheonlynonemptysubsetofR
n
thatisbothopenandclosedisR
n
itself(Exercise5.1.21),oneofAandBisR
n
andtheotherisempty.
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296 Chapter5
Real-ValuedFunctionsof
n
Variables
y
x
(3, 3)
(3, 2)
(1, 1)
(1, 2)
Figure5.1.7
IfX
1
;X
2
;:::;X
k
arepointsinR
n
andL
i
isthelinesegmentfromX
i
toX
iC1
,1i
k1,wesaythatL
1
,L
2
,...,L
k1
formapolygonalpathfromX
1
toX
k
,andthatX
1
andX
k
areconnectedbythepolygonalpath.Forexample,Figure5.1.7showsapolygonal
pathinR
2
connecting.0;0/to.3;3/. AsetSispolygonallyconnectedifeverypairof
pointsinScanbeconnectedbyapolygonalpathlyingentirelyinS.
Theorem5.1.20
AnopensetS inR
n
isconnectedifandonlyifitispolygonally
connected:
Proof
Forsufficiency,wewillshowthatifSisdisconnected,thenSisnotpolygonally
connected. LetS DA[B,whereAandBsatisfy(5.1.16). SupposethatX
1
2Aand
X
2
2B,andassumethatthereisapolygonalpathinSconnectingX
1
toX
2
. Thensome
linesegmentLinthispathmustcontainapointY
1
inAandapointY
2
inB. Theline
segment
XDtY
2
C.1t/Y
1
; 0t1;
ispartofLandthereforeinS.Nowdefine
Dsup
˚
ˇ
ˇ
tY
2
C.1t/Y
1
2A;0t1
;
andlet
X
DY
2
C.1/Y
1
:
ThenX
2
A\
B.However,sinceX
2A[Band
A\BDA\
BD;,thisisimpossible.
Therefore,theassumptionthatthereisapolygonalpathinSfromX
1
toX
2
mustbefalse.
Section5.1
Structureof
R
n
297
Fornecessity,supposethatS isaconnectedopensetandX
0
2 S. LetAbetheset
consistingofX
0
andthepointsinScanbeconnectedtoX
0
bypolygonalpathsinS.Let
BbesetofpointsinSthatcannotbeconnectedtoX
0
bypolygonalpaths.IfY
0
2S,then
Scontainsan-neighborhoodN
.Y
0
/ofY
0
,sinceSisopen.AnypointY
1
inN
.Y
0
can
beconnectedtoY
0
bythelinesegment
XDtY
1
C.1t/Y
0
; 0t1;
whichliesinN
.Y
0
/(Lemma 5.1.12)andthereforeinS. ThisimpliesthatY
0
canbe
connectedtoX
0
byapolygonalpathinSifandonlyifeverymemberofN
.Y
0
/canalso.
Thus,N
.Y
0
/AifY
0
2A,andN
.Y
0
/2BifY
0
2B.Therefore,AandBareopen.
SinceA\BD;,thisimpliesthatA\
B D
A\BD;(Exercise5.1.14). SinceAis
nonempty.X
0
2A/,itnowfollowsthatBD;,sinceifB¤;,Swouldbedisconnected
(Definition5.1.19).Therefore,ADS,whichcompletestheproofofnecessity.
WedidnotusetheassumptionthatSisopenintheproofofsufficiency.Infact,weactu-
allyprovedthatanypolygonallyconnectedset,openornot,isconnected.Theconverseis
false.Aset(notopen)maybeconnectedbutnotpolygonallyconnected(Exercise5.1.29).
OurstudyoffunctionsonRnwilldealmostlywithfunctionswhosedomainsareregions,
definednext.
Definition5.1.21
AregionSinR
n
istheunionofanopenconnectedsetwithsome,
all,ornoneofitsboundary;thus,S
0
isconnected,andeverypointofSisalimitpointof
S
0
.
Example5.1.10
Intervalsare theonlyregionsinR(Exercise 5.1.31). Then-ball
B
r
.X
0
/(Example5.1.7)isaregioninR
n
,asisitsclosure
S
r
.X
0
/.Theset
SD
˚
.x;y/
ˇ
ˇ
x
2
Cy
2
1 or x
2
Cy
2
4
(Figure5.1.8
(a)
,page298)isnotaregioninR
2
,sinceitisnotconnected. ThesetS
1
obtainedbyaddingthelinesegment
L
1
W XDt.0;2/C.1t/.0;1/; 0<t<1;
toS(Figure5.1.8
(b)
)isconnectedbutisnotaregion,sincepointsonthelinesegmentare
notlimitpointsofS
0
1
.ThesetS
2
obtainedbyaddingtoS
1
thepointsinthefirstquadrant
boundedbythecirclesx
2
Cy
2
D1andx
2
Cy
2
D4andthelinesegmentsL
1
and
L
2
W XDt.2;0/C.1t/.1;0/; 0<t<1
(Figure5.1.8
(c)
),isaregion.
MoreaboutSequencesin
RRR
n
FromDefinition5.1.13,asequencefX
r
gofpointsinR
n
convergestoalimit
Xifandonly
ifforevery>0thereisanintegerKsuchthat
jX
r
Xj< if rK:
298 Chapter5
Real-ValuedFunctionsof
n
Variables
TheR
n
definitionsofdivergence,boundedness,subsequence,andsums,differences,and
constantmultiplesofsequencesareanalogoustothosegiveninSections4.1and4.2for
thecasewherenD1.SinceR
n
isnotorderedforn>1,monotonicity,limitsinferiorand
superiorofsequencesinR
n
,anddivergenceto˙1areundefinedforn>1.Productsand
quotientsofmembersofR
n
arealsoundefinedifn>1.
L
2
L
1
(c)
(a)
L
1
(b)
y
x
y
x
y
x
Figure5.1.8
SeveraltheoremsfromSections4.1and4.2remainvalidforsequencesinR
n
,withproofs
unchanged,providedthat“j j"isinterpretedasdistanceinR
n
. (Atrivialchangeisre-
quired:thesubscriptn,usedinSections4.1and4.2toidentifythetermsofthesequence,
mustbereplaced,sincenherestandsforthedimensionofthespace.)TheseincludeThe-
orems4.1.2(uniquenessofthelimit),4.1.4(boundednessofaconvergentsequence),parts
of4.1.8(concerninglimitsofsums,differences,andconstantmultiplesofconvergentse-
quences),and4.2.2(everysubsequenceofaconvergentsequenceconvergestothelimitof
thesequence).
Section5.1
Structureof
R
n
299
5.1Exercises
WithRreplacedbyR
n
,thefollowingexercisesfromSection1:3arealsosuitableforthis
section:1.3.7-1.3.10;1.3.12-1.3.15;1.3.19;1.3.20.except
(e)
/;and1.3.21:
1.
FindaXCbY.
(a)
XD.1;2;3;1/, YD.0;1;2;0/,aD3,bD6
(b)
XD.1;1;2/, YD.0;1;3/,aD1,bD2
(c)
XD.
1
2
;
3
2
;
1
4
;
1
6
/, YD.
1
2
;1;5;
1
3
/,aD
1
2
,bD
1
6
2.
ProveTheorem5.1.2.
3.
FindjXj.
(a)
.1;2;3;1/
(b)
1
2
;
1
3
;
1
4
;
1
6
(c)
.1;2;1;3;4/
(d)
.0;1;0;1;0;1/
4.
FindjXYj.
(a)
XD.3;4;5;4/, YD.2;0;1;2/
(b)
XD.
1
2
;
1
2
;
1
4
;
1
4
/, YD.
1
3
;
1
6
;
1
6
;
1
3
/
(c)
XD.0;0;0/, YD.2;1;2/
(d)
XD.3;1;4;0;1/, YD.2;0;1;4;1/
5.
FindXY.
(a)
XD.3;4;5;4/, YD.3;0;3;3/
(b)
XD.
1
6
;
11
12
;
9
8
;
5
2
/, YD.
1
2
;
1
2
;
1
4
;
1
4
/
(c)
XD.1;2;3;1;4/, YD.1;2;1;3;4/
6.
ProveTheorem5.1.9.
7.
FindaparametricequationofthelinethroughX
0
inthedirectionofU.
(a)
X
0
D.1;2;3;1/, UD.3;4;5;4/
(b)
X
0
D.2;0;1;2;4/, UD.1;0;1;3;2/
(c)
X
0
D.
1
2
;
1
2
;
1
4
;
1
4
/, UD.
1
3
;
1
6
;
1
6
;
1
3
/
8.
SupposethatU¤0andV¤0.Completethesentence:Theequations
XDX
0
CtU; 1<t<1;
and
XDX
1
CsV; 1<s<1;
representthesamelineinR
n
ifandonlyif...
9.
FindtheequationofthelinesegmentfromX
0
toX
1
.
(a)
X
0
D.1;3;4;2/, X
1
D.2;0;1;5/
(b)
X
0
D.3;12;1;4/, X
1
D.2;0;1;4;3/
(c)
X
0
D.1;2;1/, X
1
D.0;1;1/
300 Chapter5
Real-ValuedFunctionsof
n
Variables
10.
Findsup
˚
ˇ
ˇ
N
.X
0
/S
.
(a)
X
0
D.1;2;1;3/;SDtheopen4-ballofradius7about.0;3;2;2/
(b)
X
0
D.1;2;1;3/;SD
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
jx
i
j5;1i4
(c)
X
0
D.3;
5
2
/;SDtheclosedtrianglewithvertices.2;0/,.2;2/,and.4;4/
11.
Find
(i)
@S;
(ii)
S;
(iii)
S
0
;
(iv)
exteriorofS.
(a)
SD
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
jx
i
j<3;iD1;2;3
(b)
SD
˚
.x;y;1/
ˇ
ˇ
x
2
Cy
2
1
12.
Describethefollowingsetsasopen,closed,orneither.
(a)
SD
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
jx
1
j>0;x
2
<1;x
3
¤2
(b)
SD
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
x
1
D1;x
3
¤4
(c)
SD
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
x
1
D1;3x
2
1;x
4
D5
13.
Showthattheclosureoftheopenn-ball
B
r
.X
0
/D
˚
X
ˇ
ˇ
jXX
0
j<r
istheclosedn-ball
B
r
.X
0
/D
˚
X
ˇ
ˇ
jXX
0
jr
:
14.
Prove:IfAandBareopenandA\BD;,thenA\
BD
A\BD;.
15.
Showthatiflim
r!1
X
r
exists,thenitisunique.
16.
ProveTheorem5.1.14.
17.
ProveTheorem5.1.15.
18.
Findlim
r!1
X
r
.
(a)
X
r
D
rsin
r
;cos
r
;e
r
(b)
X
r
D
1
1
r2
;log
rC1
rC2
;
1C
1
r
r
19.
Findd.S/.
(a)
SD
˚
.x;y;x/
ˇ
ˇ
jxj2;jyj1;j´2j2
(b)
SD
.x;y/
ˇ
ˇ
.x1/
2
9
C
.y2/
2
4
D1
(c)
SDthetriangleinR2withvertices.2;0/,.2;2/,and.4;4/
(d)
SD
˚
.x
1
;x
2
;:::;x
n
/
ˇ
ˇ
jx
i
jL;iD1;2;:::;n
(e)
S D
˚
.x;y;´/
ˇ
ˇ
x¤0;jyj1;´>2
20.
Provethatd.S/Dd.
S/foranysetSinR
n
.
21.
Prove:IfanonemptysubsetSofR
n
isbothopenandclosed,thenSDR
n
.
Section5.1
Structureof
R
n
301
22.
UsetheBolzano–WeierstrasstheoremtoshowthatifS
1
, S
2
, ..., , S
m
, ... . is s an
infinitesequenceofnonemptycompactsetsandS
1
S
2
S
m
,then
T
1
mD1
S
m
isnonempty. Showthattheconclusiondoesnotfollowifthesetsare
assumedtobeclosedratherthancompact.
23.
SupposethatasequenceU
1
,U
2
,... ofopensetscoversacompactsetS. Without
usingtheHeine–Boreltheorem, showthatS 
S
N
mD1
U
m
forsomeN. H
INT
:
ApplyExercise5.1.22tothesetsS
n
DS\
S
n
mD1
U
m
c
:
(ThisisaseeminglyrestrictedversionoftheHeine–Boreltheorem,validforthe
casewherethecoveringcollectionHisdenumerable. However,itcanbeshown
thatthereisnolossofgeneralityinassumingthis.)
24.
ThedistancefromapointX
0
toanonemptysetSisdefinedby
dist.X
0
;S/Dinf
˚
jXX
0
j
ˇ
ˇ
X2S
:
(a)
Prove:IfSisclosedandX
0
2R
n
,thereisapoint
XinSsuchthat
j
XX
0
jDdist.X
0
;S/:
H
INT
:ApplyExercise5.1.22tothesets
C
m
D
˚
X
ˇ
ˇ
X2SandjXX
0
jdist.X
0
;S/C1=m
; m1:
(b)
ShowthatifSisclosedandX
0
62S,thendist.X
0
;S/>0.
(c)
Showthattheconclusionsof
(a)
and
(b)
mayfailtoholdifSisnotclosed.
25.
ThedistancebetweentwononemptysetsSandT isdefinedby
dist.S;T/Dinf
˚
jXYj
ˇ
ˇ
X2S;Y2T
:
(a)
Prove:IfSisclosedandT iscompact,therearepoints
XinS and
YinT
suchthat
j
X
YjDdist.S;T/:
H
INT
:UseExercises5.1.22and5.1.24:
(b)
Undertheassumptionsof
(a)
,showthatdist.S;T/>0ifS \T D;.
(c)
Showthattheconclusionsof
(a)
and
(b)
mayfailtoholdifSorT isnot
closedorT isunbounded.
26. (a)
Prove: IfacompactsetSiscontainedinanopensetU,thereisapositive
numberrsuchthattheset
S
r
D
˚
X
ˇ
ˇ
dist.X;S/r
iscontainedinU.(YouwillneedExercise5.1.24here.)
(b)
ShowthatS
r
iscompact.
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