8-11

==== Question 8.11. ====
If f_{k} \rightarrow f  in L^{p}, 1 \leq p<\infty, g_{k} \rightarrow g pointwise, and \left\|g_{k}\right\|_{\infty} \leq M  for all k, prove that f_{k} g_{k} \rightarrow f g  in L^{p}.
==== Solution. ====
By Minkowski's inequality,
\left\|f_{k} g_{k}-f g\right\|_{p} \leq\left\|f_{k} g_{k}-f g_{k}\right\|_{p}+\left\|f g_{k}-f g\right\|_{p} \leq M\left\|f_{k}-f\right\|_{p}+\left\|f g_{k}-f g\right\|_{p} .
Since g_{k} \rightarrow g  pointwise and since \left|g_{k}\right| \leq M  a.e., it follows that |g| \leq M  a.e., so \left|g_{k}-g\right| \leq 2 M  a.e. Thus,
\int\left|f g_{k}-f g\right|^{p} \leq \int(2 M)^{p}|f|^{p}<\infty,
so \left|f g_{k}-f g\right|^{p} \in L^{1} . By the dominated convergence theorem, we have
\lim _{k \rightarrow \infty} \int\left|f g_{k}-f g\right|^{p}=\int \lim _{k \rightarrow \infty}|f|^{p}\left|g_{k}-g\right|^{p}=0,
so \left\|f g_{k}-f g\right\|_{p} \rightarrow 0 . 
Since we also have \left\|f_{k}-f\right\|_{p} \rightarrow 0 , we can write
\lim _{k \rightarrow \infty}\left\|f_{k} g_{k}-f g\right\|_{p}=0.