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Applications
45
numbers
DE(V(J))
and
DE(V(t))
arefinite.Inparticular,thefunction
v
iswell-defined.
Thethirdremarkismethodologicalinnature.Aswewillseebelow,
thewholeproofcanbedividedintosmallpieces,mostofwhichare
establishedusingthefollowingapproach:first,usethedefinitionsof
equicontinuityandmeasureofnon-compactnesstofindsomesets,
then,usingthesesets,defineacovering,andfinally,estimatethe
diameterofeachsetbelongingtothecovering.
Lastly,letuswarnthereaderthatthisisquiteaninvolvedproof,
andwearegoingtoneedalotoflittlevariableshereandthere–and
hencewewillsometimesreusethesamelettermorethanonce,oruse
similarsymbolsforthingsbelongingtodifferentsets.Wetriedhard
toavoidsuchtrapsinthisbook,butthisisoneoftheplaceswhenwe
hadtothrowinthetowel.
(a)
Now,weturntoprovingthatthefunction
v
isuniformlycon-
tinuouson
J
.Foranygiven
8>
0,let
δ:=δ(1
48)
bechosenasin
Definition1.2.4ofequicontinuityof
V
,andlet
S
,
t∈J
besuchthat
|tlS|<δ
.Moreover,let
V1
,
...
,
Vn⊆E
besetsformingafinitecover-
ingof
V(t)
withdiametersnotgreaterthan
DE(V(t))+
1
48
.Forevery
i∈{
1,
...
,
n}
letusdefine
Ui:={u∈E|inf
w∈Vi
"ulw"<1
48}
.Observe
that
V(S)⊆U1∪...∪Un
.Indeed,if
u∈V(S)
,then
u=w(S)
forsome
w∈V
.Since
"w(S)lw(t)"<1
48
and
w(t)∈Vj
forsome
j∈{
1,
...
,
n}
,
weinferthatu∈Uj.
Now,wewillestimatethediametersofthesets
Ui
.Fix
i∈{
1,
...
,
n}
.
Forevery
u1
,
u2∈Ui
,bythedefinitionofinfimum,itispossibletofind
w1
,
w2∈Vi
suchthat
"u1lw1"<
3
88
and
"u2lw2"<
3
88
.Hence,
"u1lu2"<"u1lw1"+"w1lw2"+"w2lu2"<DE(V(t))+8
,
andtherefore
diamUi<DE(V(t))+8
.Consequently,
DE(V(S))<
DE(V(t))+8
.Likewise,
DE(V(t))<DE(V(S))+8
.Thisprovesthatthe
functionvisuniformlycontinuousonJ.
(b)
Theproofisdividedintotwoparts.Webeginwithshowing
that
DC(V)=DE(V(J))
.Let
8>
0andlet
V1
,
...
,
Vn⊆C(J
;
E)
bea
finitecoveringof
V
suchthat
diamVi<DC(V)+
1
28
for
i=
1,
...
,
n
.
n
Replacing
Vi
by
V∩Vi
,wemayclearlyassumethat
V=
U
Vi
.More-
i=1
1
