proof of composition limit law for uniform convergence
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|
|Date of creation||2013-03-22 15:23:08|
|Last modified on||2013-03-22 15:23:08|
|Last modified by||stevecheng (10074)|