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