Weak-strong product convergence lemma
= Weak-strong product convergence lemma
If $f_n\rightharpoonup f$ in $L^p$, $g_n\to g$ in $L^q$, and $1/p+1/q=1$, then
$$
\int f_ng_n\longrightarrow\int fg.
$$
Indeed, split the difference into the weak pairing with $g$ and a term bounded by the <Holder inequality> using $g_n-g$.