Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 22F / c / i

First suppose $1\le p<\infty$. Choose a subsequence so thatBy Tonelli theorem,Hence the sum inside the integral is finite almost everywhere, which forces $|f_{n_k}(x)-f(x)|\to 0$ almost everywhere.

If $p=\infty$, choose representatives and remove the countable union of the null sets on whichOutside that null set, norm convergence bounds the pointwise difference and the entire sequence converges to $f$. Thus in every case an almost-everywhere convergent subsequence exists, as stated in Almost-everywhere convergent subsequence from Lp convergence.