proof of composition limit law for uniform convergence

Theorem 1.

Let X,Y,Z be metric spaces, with X compactPlanetmathPlanetmath and Y locally compact. If fn:XY is a sequence of functions converging uniformly to a continuous functionMathworldPlanetmathPlanetmath f:XY, and h:YZ is continuous, then hfn convergePlanetmathPlanetmath to hf uniformly.


Let K denote the compact set f(X)Y. By local compactness of Y,for each point yK, there is an open neighbourhood Uy of y such that Uy¯ is compact. The neighbourhoods Uy cover K, so there is a finite subcover Uy1,,Uyn covering K. Let U=iUyiK. Evidently U¯=iUyi¯ is compact.

Next, let V be the δ0-neighbourhood of K contained in U, for some δ0>0. V¯ is compact, since it is contained in U¯.

Now let ϵ>0 be given. h is uniformly continuousPlanetmathPlanetmath on V¯, so there exists a δ>0 such that when y,yV¯ and d(y,y)<δ, we have d(g(y),g(y))<ϵ.

From the uniform convergenceMathworldPlanetmath of fn, choose N so that when nN, d(fn(x),f(x))<min(δ,δ0) for all xX. Since f(x)K, it follows that fn(x) is inside the δ0-neighbourhood of K, i.e. both y=fn(x) and y=f(x) are both in V. Thus d(g(fn(x)),g(f(x)))<ϵ when nN, uniformly for all xX. ∎

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