# How to compute the limit of this integral?

Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that $f(0)\neq f(1)$. Define $f_n(x)=f(x^n)$ then I want to prove that $\lim_{n\rightarrow\infty}\int^1_0 f_n(x)dx=f(0)$.

Now, you can easily prove that the convergence is not uniform, so we can’t switch the sign of limit and the sign of the integral. I tried to do a bunch of things to solve this problem, for example if $S(f_n,\sigma_N)$ is the upper sum of $f_n$ with respect to the equispaced partition $\sigma_N$ then i tried to prove that $\lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty}S(f_n,\sigma_n)=f(0)$, this is true if we can switch the limits, but I don’t know why we can. Could you help me please?

A large interval where you can bound the value of the function to be near $f(0)$ for large $n$.