 Almost-everywhere convergent subsequence from Lp convergence (source code)

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= Almost-everywhere convergent subsequence from Lp convergence
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If $f_n\to f$ in $L^p$ for $1\leq p<\infty$, choose a subsequence with $\lVert f_{n_k}-f\rVert_p^p\leq2^{-k}$. 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.

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