Thread: 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

Your sequence is uniformly bounded because is. Also the uniform boundedness of the can be used to check that is equicontinuous and then conclude by Arzela Ascoli.