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

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

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