Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function.
- Show that for each $\epsilon\in(0,1)$, $\lim\limits_{n\rightarrow\infty}\int\limits_0^{1-\epsilon}f(x^n)dx=(1-\epsilon)f(0)$
- Find $\lim\limits_{n\rightarrow\infty}\int\limits_{0}^{1}f(x^n)dx$.
Hint: Start by explaining why $f$ is bounded.
For my Answer:,
I have done the part one by using the fact that
\begin{align*}
\left|\int\limits_{0}^{1-\epsilon}f(x^n)dx-(1-\epsilon)f(0)\right|\leq \int\limits_0^{1-\epsilon}|f(x^n)-f(0)|dx
\end{align*}
And with the continuity of $f$ at zero along with $x^n\leq(1-\epsilon)^n\rightarrow0$
But for the part two what I can see is that:
\begin{align*}
\lim\limits_{n\rightarrow\infty}\int\limits_{0}^{1}f(x^n)dx &=\lim\limits_{n\rightarrow\infty}\lim\limits_{\epsilon\rightarrow0}\int\limits_{0}^{1-\epsilon}f(x^n)dx
\end{align*}
But there after if I need to make use of part (1) then I have to interchange the limits. So is it possible. If so, I would like to know what are the conditions we need to have for such an interchange.
Moreover I would like a feedback on the hint given. (Why is it given?)