Let $f \in \mathcal{S}(\mathbb{R})$ and $T_n$ be a sequence of tempered distributions converging to $T$, which is also a tempered distribution. Is it true that $fT_n \to fT$?
My solution to this seems quite trivial so I'm not sure if there is something I'm missing, or if it's just a trivial question. What I did is $$\lim_{n\to\infty} <fT_n, \phi> = \lim_{n\to\infty} <T_n, f\phi> = <T,f\phi> = <fT,\phi>$$ And thus, $fT_n \to fT$.
Am I missing something?