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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.

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    Thank for the replies, I was of the impression it was 'at least one' but 'ONE' was emphasized so I'm still confused as to what it meant. Also, Jose could you expand a bit on the application of a diagonal argument in this particular case?2012-11-26

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Hint: Apply Arzelà–Ascoli theorem to the sequence $\int_0^xf_n(t)dt$, $x\in[0,1]$.

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    @jimmywho: You are welcome.2012-11-27