Uniform Convergence on an Equicontinuous Set

[Kreizhn]

Homework Statement

Let (X,d) be a compact metric space, (f_n) be an equicontinuous sequences of functions in C(X, \mathbb{R})such that, for every fixed x in X, (f_n(x)) \to 0.

Show that (f_n) converges uniformly to the zero function

Homework Equations

The Attempt at a Solution

First things first, I started by noting that since X is compact, then C(X,R) (the space of all continuous functions from X to R) is equivalent to C_b(X,R), the set of all continuous bounded functions from X to R. Now R complete implies that F_b(X,R) is complete (set of bounded functions), and C_b(X,R) is a closed subset of F_b(X,R) and so is also complete. Thus it suffices to show that (f_n) is Cauchy with respect to the distance induced by the uniform norm. That is

\rho(f,g) = \sup \{ |f(x) - g(x) | | x \in X \}

Let \epsilon > 0. Then (f_n) uniformly continuous implies \exists \delta >0 such that \forall x,y \text{ satisfying } d(x,y) < \delta, |f(x)-f(y)| < \epsilon. Since (f_n(x)) \to 0, \exists n_x for each fixed x, such that |f_n(x)| < \epsilon.

Since X is compact, let x_1, \ldots, x_p be a delta-net for X. Then \forall x \in X, \exists x_j \text{ such that } d(x,x_j) < \delta and so |f_n(x) - f_n(x_j)| < \epsilon for all n.

I'm not sure if I ended up doing some of this for no reason, but I can't quite figure out where to go from here.

 

[Kreizhn]

Anyone?

 

[Dick]

You know f_n converges to zero. That means that for every e>0 there is a N(x) such that for all n>N(x) |f_n(x)|<e. You only have to show that it's uniform. If f_n is continuous that means that there is a d_n(x) such that if |x-x'|<d_n(x) then |f_n(x)-f_n(x')|<e. Equicontinuity let's you eliminate the dependence of d on n. Now you want to eliminate the dependence on x. Use compactness.