Almost-everywhere convergent subsequence from Lp convergence

If $f_n\to f$ in $L^p$ for $1\le p<\infty$, choose a subsequence with $\Vert f_{n_k}-f{\Vert }_p^p\le 2^{-k}$. Tonelli's theorem makes ${\sum }_k|f_{n_k}-f{|}^p$ finite almost everywhere, so $f_{n_k}\to f$ almost everywhere. For $p=\infty$, norm convergence itself gives almost-everywhere convergence after removing one common null set.