# Math Help - prove existence of a subsequence that converges uniformly

1. ## 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

2. 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.