Q:

Prove that the integral of a sequence converges uniformly

I've been stuck on this problem for a bit now:
Let $g$ and $f_0$ be continuous functions on $[0,1]$. Define the sequence on $[0,1]$ by
$$f_n(x) = \int_0^t g(t)f_{n-1}dt.$$
I have to prove that the sequence is converges uniformly. 
Here are my thoughts: I know $f_n$'s are continuous and I also tried to solve this question using the $\epsilon-\delta$ definition for continuity and uniform convergence without any luck. 
Any hint? 

A:

I suppose that the upper limit in the integral is $x$ and not $t$.
Let $G$ and $F$ be constants such that $|g(x)|\le G$ and $|f_0(x)|\le F$ for all $x\in[0,1]$. Then
$$\begin{align}
|f_1(x)|&\le\int_0^xG\,F\,dt=G\,F\,x\\
|f_2(x)|&\le\int_0^xG^2\,F\,t\,dt=\frac{G^2\,F}{2}\,x^2\\
|f_3(x)|&\le\int_0^x\frac{G^3\,F}{2}\,t^2\,dt=\frac{G^3\,F}{2\cdot3}\,x^3\\
\end{align}$$
Can you see a pattern?