Math Help - Uniform Convergence

1. Uniform Convergence

I have been staring at this problem all day and I can't figure how to even begin to form subsequence. Can someone give me a jump start?

Let $\{f_n\}$ be a uniformly bounded sequence of functions which are Riemann-integrable on [a, b], and put

$F_n(x) = \int_a^x f_n(x), dt (a \leq x \leq b).$

Prove that there exists a subsequence $\{F_{n_k}\}$ which converges uniformly on [a, b].

2. The problem is from Blue Rudin chapter 7 problem 18. Also I believe I am supposed to follow a format similar to the proof of Theorem 7.23. I hope this helps.