the uniform convergence of the sequence of functions Let $f_1:[a,b]\rightarrow \mathbb{R}$ be a Riemann integrable function. Define the sequence of functions $f_n:[a,b] \rightarrow \mathbb{R}$ by
$f_{n+1}(x)=\int_a^x f_n(t)dt,$
for each $n\ge 1$ and each $x\in [a,b]$. Prove that the sequence of functions 
$g_n(x)=\sum\limits_{k=1}^n f_k(x)$
converges uniformly on [a,b]. 
I try to use Ascoli Theorem to prove this problem by considering 
$|g_n(x)-gn(y)|\le\sum\limits_{k=1}^n |f_k(x)-f_k(y)|$. However, $|f_k(x)-f_k(y)|\le M |b-a|^{k-2} |x-y|, $ where $M>0$ is a constant achieving by the condition that $f_1$ is an integrable function. However, this estimation depends on $k$, so I cannot apply Ascoli-Azela Theorem. 
Thank you in advance for your help.

Hint: 
$$|f_1(x)| \leqslant M, \\|f_2(x)| \leqslant M(x-a), \\ |f_3(x)| \leqslant \frac{1}{2!}M(x-a)^2, \\ \ldots$$
