Let K be a nonempty compact set in ℝ^{m} and consider all the continuous functions defined on this set
having values in ℝ^{n}. It is desired to give conditions which will show that a subset of C
(K, ℝn)
is compact.
First is an important observation about compact sets.
Proposition 7.2.1Let K be a nonempty compact subset of ℝ^{m}. Then for each ε > 0 there is afinite set of points
{xi}
_{i=1}^{r}such that K ⊆∪_{i}B
(xi,ε)
. This finite set of points is called an ε net.If D_{1∕k}is this finite set of points corresponding to ε = 1∕k, then ∪_{k}D_{1∕k}is a dense countable subsetof K.
Proof:The last claim is obvious. Indeed, if B
(x,r)
≡
{y ∈ K : |y − x | < r}
, then consider D_{1∕k}
where
1
k
<
1
3
r. Then the given ball must contain a point of D_{1∕k} since its center is within 1∕k of some point
of D_{k}. Now consider the first claime about the ε net. Pick x_{1}∈ K. If B
(x1,ε)
⊇ K, stop. You have your ε
net. Otherwise pick x_{2}
∕∈
B
(x1,ε)
. If K ⊆ B
(x1,ε)
∪B
(x2,ε)
, stop. You have found your ε net. Continue
this way. Eventually, the process must stop since otherwise, you would have an infinite sequence of
points with not limit point because they are all ε apart. This contradicts the compactness of K.
■
_{k=1}^{∞} is a Cauchy sequence and so it converges to some
g
(x )
. This shows the claim.
It remains to verify that g is continuous. This is implied if we can show uniform convergence of g_{k}
and thus is obtained from the following claim which is a general result about equicontinuous
functions.
Claim:If
{gk}
_{k=1}^{∞} is equicontinuous and converges pointwise to g on a compact set K, then it
converges uniformly on K.
Proof of claim: Let ε > 0 be given and let δ go with ε∕4 in the definition of equicontinuous. By
compactness, there are finitely many points of K
{x1,⋅⋅⋅,xs}
such that K ⊆∪_{i=1}^{s}B
(xi,δ)
. There exists
N_{i} such that if k,l ≥ N_{i}, then
|gl(xi)− gk(xi)|
<
ε4
. Thus if N ≥ max
{Ni,i = 1,⋅⋅⋅,s}
, then for all
x_{i},
|gl(xi)− gk (xi)|
<
ε4
if k ≥ N. Then for k,l ≥ N, and x arbitrary, let x ∈ B