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Let $\{f_n\}$ be an equicontinuous family of functions on $[0,1]$ such that each $f_n$ is pointwise bounded and $\int_{[a,b]} f_n(x)dx \rightarrow 0$ as $n\rightarrow \infty$, for every $ 0\leq a \leq b \leq 1$.

Show $f_n$ converges to $0$ uniformly.

For this question I know that there exists a uniformly convergent subsequence $f_{n_k}$ by Arzela-Ascoli Theorem. For this uniformly convergent sequence I know $$\lim_{n\rightarrow \infty} \int_a^bf_{n_k}(x)dx = \int_a^b \lim_{n\rightarrow \infty}f_{n_k}(x)dx$$ Since the left side is zero If we assume $\lim_{n\rightarrow \infty}f_{n_k}(x)dx \neq \ 0$ for some $x\in [0,1]$ Then uniform continuity of the limit implies that it is $\neq 0$ on some interval which implies the integral is $\neq 0$ Which is a contradiction . Thus $f_{n_k}$ must converge uniformly to $0$. I don't see how to get to $f_n$ converges uniformly to $0$ though.

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    http://www.tricki.org/article/Prove_that_a_sequence_converges_by_passing_to_a_subsequence2011-07-12
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    @Theo: Is that really helpful? The technique you use to prove this is not in that list if I understood it correctly.2011-07-12
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    @Jonas: I'm using the first bullet point: Every subsequence has a subsubsequence which converges to zero. I'm providing the missing argument.2011-07-12

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