Sequence with bounded l2 norm

Question

Let $f_i \colon [0,1]\rightarrow \mathbb{R}$ be a sequence of functions which converge to $f_\infty$ pointwise. How can I prove that $\lim_{i\rightarrow \infty} \int f_i d\lambda= \int f_\infty d\lambda$, when $||f_i||_2\leq 1$?

My attemp is to use Egorov and then try to use the bound on the $l_2$-norm in the set with small measure, but I don't know how to prove that $||f_\infty||_2$ is not infinite.

Answer 1

Say $||f_n||_2\le1$, $f_n\to f$ almost everywhere, $f_n\to f$ uniformly on $[0,1]\setminus E$, and $m(E)<\epsilon$. Cauchy-Schwarz shows that $\int_E|f_n|\le\epsilon^{1/2}$. And then Fatou's Lemma shows that $\int_E|f|\le\epsilon^{1/2}$.