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Continuous Compact Point Convergent Sequence Neighborhood Metric Space Finite Function Superset Sequence Site Open Question Uniform Convergence Locally Compact Cover Uniformly Continuous
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Proof Of Composition Limit Law For Uniform Convergence
Theorem 1
Let 
be metric spaces, with 
compact
and 
locally compact.
If
is a sequence
of functions
converging uniformly
to a continuous function
, and
is continuous, then
converge
to 
uniformly.
be metric spaces, with compact
and locally compact.
If
is a sequence
of functions
converging uniformly
to a continuous function
, and
is continuous, then
converge
to uniformly.
Proof.
Let 
denote the compact set
. By local compactness
of 
,for each point
, there is an open
neighbourhood
of 
such that
is compact. The neighbourhoods 
cover
, so there is a finite
subcover
covering
. Let
. Evidently
is compact.
denote the compact set
. By local compactness
of ,for each point
, there is an open
neighbourhood
of such that
is compact. The neighbourhoods cover
, so there is a finite
subcover
covering
. Let
. Evidently
is compact.
Next, let 
be the 
-neighbourhood of 
contained
in 
, for some
.
is compact, since it is contained in
.
Now let
be given.
is uniformly continuous
on
, so
there exists a
such that
when
and
,
we have
.
From the uniform convergence
of 
, choose 
so that
when 
,
for all 
.
Since
, it follows that 
is inside the
-neighbourhood of 
, i.e. both
and 
are both in 
. Thus
when 
,
uniformly for all 
.
