# convergence in $L^2$

Let $(f_n)$ be a sequence of measurable functions on $[0,1]$ such that $\int_{0}^1 |f_n|^2 dm \le C$.

Assume that there exist a function $f$ such that $\int_{0}^1 |f_n-f|dm \to 0$ as $n \to \infty$

show that $\int_{0}^1 |f|^2 dm \le C$ and it is true that $\int_{0}^1 |f_n-f|^2dm \to 0$ as $n \to \infty$ ?

I have no idea how to begin, any hints to start me off would be appreciated. Thanks

-
The conclusion is false. Please check if your question is correctly formulated. – 23rd Dec 8 '12 at 16:32
@ richard, got the mistake, thanks – jake Dec 8 '12 at 17:10
You are welcome! – 23rd Dec 8 '12 at 17:22

Begin extracting a subsequence $\{f_{n_k}\}$ such that for all $k$, $$\int_{[0,1]}|f_{n_k}-f|dm< 4^{-k},$$ then that the sequence $\{f_{n_k}\}$ converges almost everywhere to $f$.
The second one is not true: take $f_n:=\sqrt n\chi_{(0,n^{-1})}$, whose $L^2$ norm is $1$, and $f=0$. Then $\int_{(0,1)}f_n=n^{-1/2}\to 0$.