Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 1 / 22F / c / Solution

For $1<p<\infty$, apply Fatou lemma to the nonnegative sequence $|f_n{|}^p$, which converges pointwise to $|f{|}^p$:For $p=\infty$, each inequality $\Vert f_n{\Vert }_{\infty }\le M$ fails pointwise only on a null set. Outside the union of these countably many null sets, $|f_n(x)|\le M$ for every $n$, so the pointwise limit also satisfies $|f(x)|\le M$. Therefore in both cases