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Applications
39
sequenceofnon-emptyclosedsetsinacompletemetricspace.Moreover,
∞
assumethatlim
n→∞
diamAn=0.Then,
n=1
Π
Anisasingleton.
Proof.
As
D(An)<diamAn
for
n∈N
,Theorem1.1.32assuresus
∞
that
A:=
Π
An
isnon-empty.Since
A⊆An
forevery
n∈N
,we
n=1
have
diamA<diamAn
forall
n∈N
,andhence
diamA=
0,which
meansthatAcannotcontaintwodistinctpoints.
ItmaycomeasasurprisethattheCantortheoremcanbeusedto
characterizecompletenessofametricspace.
1010340Proposition0
Ifforeverynon-increasingsequence
(An)n∈N
ofnon-
emptyclosedsubsetsofametricspace
X
suchthat
n→∞
lim
diamAn=
0the
∞
intersection
Π
Anisnon-empty,thenthespaceXiscomplete.
n=1
Proof.
Let
(xk)k∈N
beaCauchysequenceofelementsof
X
.Define
An:={xk|k>n}
for
n∈N
.Clearly,thesets
An
arenon-empty,
closedandformanon-increasingsequence.Moreover,
diamAn→
0
as
n→+∞
.Indeed,forevery
8>
0thereexistsanindex
n0
such
that
d(xk
,
xl)<8
forall
k
,
l>n0
.Thus,
diamAn<8
for
n>n0
(cf.Exercise1.3.6(c)),whichshowsourclaim.This,bytheassumption,
∞
∞
meansthattheintersection
Π
An
isnon-empty.Let
x∈
Π
An
.To
n=1
n=1
seethat
x
isthelimitofthesequence
(xk)k∈N
(andhencethemetric
space
X
iscomplete),itsufficestonotethat
d(x
,
xk)<diamAk
for
everyk∈N.
1
102Applications
Intheprevioussectionwediscussedbasicpropertiesofmeasuresof
non-compactness.Now,wewouldliketopresenthowthetechniques
wedevelopedcanbeappliedtoabstractordinarydifferentialequations.
(Here,flabstract”meansthatwewillbeworkingwithfunctionstaking
valuesinanarbitraryBanachspaceandequationsinvolvingsuch
functions.)
