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 be a uniformly bounded sequence of functions which are Riemann-integrable on [a, b], and put



Prove that there exists a subsequence which converges uniformly on [a, b].

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.

Thank you for any advice.