Show that $\int\left\lvert f_n\right\rvert\,d\lambda\to\int\left\lvert f\right\rvert\,d\lambda\implies\int\left\lvert f_n-f\right\rvert\,d\lambda\to0$
If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$