- #1

- 3,149

- 8

## Homework Statement

So, as the title suggests, I'd like to check a few ideas, since I didn't yet complete the proof.

**Theorem.**Let X be compact, and let fn be a sequence of functions in C(X, R^k). If the collection {fn} is pointwise bounded and equicontinuous, then the sequence fn has a uniformly convergent subsequence.

## The Attempt at a Solution

First, I am trying a proof for k = 1, I'll comment on the proof of the other cases after I prove this.

Let x be in X. Then the sequence {fn(x) : n is in N} is bounded, and hence has a convergent subsequence, whose limit we'll deonte with Lx, so fn(x) --> Lx. Define the function f : X --> R with f(x) = Lx. I see a way to prove that fn --> f uniformly, but to prove this, I'd need to prove that f is continuous.

Of course, there's no reason to believe it is, although I got a hunch (probably wrong).

And of course, I know I need not prove continuity of a function to which fn converges uniformly, since this function must be continuous, by the uniform limit theorem.

But the concept of the proof probably includes an idea of how to define a specific function to which fn converges uniformly.

I don't require any hints yet, but I only want to see if there is something to this idea I've lain out above (i.e. defining f in this specific way, although by standard means I can't show continuity, since the condition depends on the integers related to convergence of the functions fn)