Problem: Let $(f_n)$ be a uniformly bounded sequence of real valued continuous functions on $[0,1]$. Prove that there is ONE subsequence $(f_{n_k})$ such that for every $0\le a < b \le 1$, we have $\lim_{k\to\infty} \int_a^b \! f_{n_k}(t) dt $ exists.
Context: Advanced Undergraduate Analysis. Familiar with Real Analysis by Carothers and Principles of Analysis by Rudin
I think it would be obvious to show this for all rationals inbetween a and b but I do not know how to start showing that there is a single subsequence. Any help would be appreciated, Thank you.