If $f_n$ and $f$ are integrable, $f_n \to f$ a.e., and $\int |f_n| \to \int |f|$, then $\int |f_n-f| \to 0$.
How can I prove this? The limit definition proof involving $N$ and $\epsilon$ is not working for me ... I can't get a certain triangle inequality to work. But is this kind of proof the way to arrive at the result?
