About convergence in L^1

Question

Let $(f_n)_{n \in \mathbb{N}} \subset L^1$ and $f \in L^1$ such $f_n \longrightarrow f \ a.e. $ and $ ||f_n||_1 \longrightarrow ||f||_1 $; then $ ||f_n - f||_1 \longrightarrow 0$. Why?