pointwise convergence with integrals

Question

Let $(f_n)$ be a sequence of uniformly bounded continuous functions on $[0,1]$ that converges pointwise to a continuous function $f$. Then $\int f_n\to\int f$. Does anyone know a proof of this?

Answer 1

By Egorov Theorem, denoting that $I=[0,1]$, for $\epsilon>0$, there exists some $F$ such that $|I-F|<\epsilon$ and $f_{n}\rightarrow f$ uniformly on $F$.

We have \begin{align*} \int_{I}|f_{n}-f|&=\int_{F}|f_{n}-f|+\int_{I-F}|f_{n}-f|\\ &\leq\int_{F}|f_{n}-f|+\int_{I-F}\left(\sup_{n}\|f_{n}\|_{\infty}+\|f\|_{\infty}\right)\\ &\leq\int_{F}|f_{n}-f|+\epsilon\left(\sup_{n}\|f_{n}\|_{\infty}+\|f\|_{\infty}\right), \end{align*} and we know that $\displaystyle\int_{F}|f_{n}-f|\rightarrow 0$ by the uniform convergence.