Problem: If $f_0$ is a continuous function in $[0,a]$, $a>0$, show that the sequence $\{f_k\}$ (on that same interval) defined the recursive relatio n$f_k(x)=\displaystyle\int_0^x f_{k-1}(t)dt$ converges uniformly to the null function.
Some osbervations I could figure out, are:
- If $f_0$ is continuos, integrating it will result in a continuous fuction.
- I'm not sure about this one, I think we need $f_k\geq 0,$ but every term of the sequence is bounded by $\int_0^x f_{k-1}(t)dt\leq\int_0^a f_k(t)dt$, being the last one a number.
I also tried to take $\lim\limits_{k\to\infty}\sup\limits_{x\in[a,b]}|f_k(x)-f(x)|$ as if it goes to $0$ thesis is true.
When I tried to take the pointwise convergence, I write $\lim\limits_{k\to\infty}f_k(x)=\lim\limits_{k\to\infty}\int_0^x f_{k-1}(t)dt$.
There might by way using derivatives and theoremes related to them and uniformly convergence, but couldn't figure it out.
Appreciate the help.