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.
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.
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 .
∎
Title | proof of composition limit law for uniform convergence |
---|---|
Canonical name | ProofOfCompositionLimitLawForUniformConvergence |
Date of creation | 2013-03-22 15:23:08 |
Last modified on | 2013-03-22 15:23:08 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 4 |
Author | stevecheng (10074) |
Entry type | Proof |
Classification | msc 40A30 |