# Another Convergence Question

1. Jan 25, 2009

### e(ho0n3

The problem statement, all variables and given/known data
Let $$\{f_n\}$$ be a uniformly bounded sequence of functions which are Riemann-integrable on [a,b]. Let
$$F_n(x) = \int_a^x f_n(t) \, dt$$
Prove that there exists a subsequence of $$\{F_n\}$$ which converges uniformly on [a,b].

The attempt at a solution
I was thinking, since $$\{f_n\}$$ is uniformly bounded, there is an M such that $$F_n(x) \le M(x - a) \le M(b - a)$$ for all n, for all x. Now this automatically means that $$\{F_n\}$$ converges uniformly right? But then if $$\{F_n\}$$ converges uniformly, why is the problem requesting for a subsequence? I must have done something wrong.

2. Jan 25, 2009

### Dick

The f_n aren't given to be convergent. Sure the F_n are bounded, but that certainly doesn't mean they are convergent. Can't you think of a theorem that guarantees the existence of a uniformly convergent subsequence in a family of functions? What are it's premises?

3. Jan 25, 2009

### e(ho0n3

You mean the Arzela-Ascoli Theorem right? The Wikipedia article actually states what I'm trying to prove I think:

the hypotheses being that the sequence must be uniformly bounded and equicontinuous. I have already shown that the sequence is uniformly bounded right? So all I need to show is that it is equicontinuous. I think I can handle that.

4. Jan 25, 2009

### Dick

You've got it.