Hello,

I want to show that the family of functions defined by $f_n(x)=\int_0^x f_{n-1}(t) dt$ converges uniformly on $[0,a]$.

Since I don't need a particular limit I've been fishing around with proof by contradiction.

I know a fair amount about the family:

Every member is continuous, so uniformly continuous as they are defined on a compact set (this should mean that the family is equicontinuous under Rudins definition in chapter 7.)

Every member is bounded, so the family is uniformly bounded.

Each $f_n$ is integrable so the $\int_0^x f_{n-1}(t) dt$ is finite at every $x\in [0,a]$.

I can use the ArzeląAscoli theorem to show the existence of a uniformly convergent subsequence (I wanted to use this condition to get a contradiction)

I can use the mean value theorem to show that $|f_n(x)-f_m(x)|=|f_n(c)x-f_m(c')(x)|\leq |f_n(c)-f_m(c')||a|= |\int_0^c f_{n-1}(t) dt - \int_0^{c'} f_{m-1}(t)dt|$

I should have $f_n(x) = \int_0^x f_{n-1}(t)dt=F(x)-F(0)$ as well no?

I'm not so sure what else to do at the moment. I'll try again in the morning.