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Math Help - prove existence of a subsequence that converges uniformly

  1. #1
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    Cool prove existence of a subsequence that converges uniformly

    Let fn be a uniformly bounded sequence of integrable functions on [a,b] (not necessarily continuous). Set

    Fn(x) = integral (x to a) fn(t)dt

    for x is an element in [a,b].


    Prove that there exists a subsequence of the sequence (Fn) that converges absolutely uniformly on [a,b].

    any help would be appreciated
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  2. #2
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    Your sequence (F_n)_n is uniformly bounded because f_n is. Also the uniform boundedness of the f_n can be used to check that (F_n)_n is equicontinuous (|F_n(x)-F_n(y)|\leq \sup_n f_n |x-y|) and then conclude by Arzela Ascoli.
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