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.
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