If $f_n\to f$ in $L^p$ and $g_n$ is uniformly bounded and $g_n\to g$ a.e. then $g_nf_n\to gf$ in $L^p$.

Question

For $1\leqslant p<\infty$, let $\langle f_n\rangle $ be a sequence of functions in $L^p$ that converges in $L^p$-norm to a function $f\in L^p$. Let $\langle g_n\rangle$ be a sequence of measurable functions such that $|g_n|\leqslant M$ for all $n\in\mathbb N$ and $g_n\to g$ a.e. Prove that $g_nf_n$ converges to $gf$ in $L^p$.

I'm so lost how to even start this question! So, please help me!

Answer 1

Observe that $$\|f_ng_n-fg\|_p=\|f_ng_n-fg_n+fg_n-fg\|_p\leq\|(f_n-f)g_n\|_p+\|f(g_n-g)\|_p.$$ Note that $$\|(f_n-f)g_n\|_p\leq M\|f_n-f\|_p\to 0.$$ And $|f(g_n-g)|\leq 2M|f|$, thus dominated convergence theorem yields $$\|f(g_n-g)\|_p\to 0.$$