Show $\mathcal{D}=C_c^\infty(\mathbb R^n)$ is dense in the Schwartz space $\mathcal{S}(\mathbb R^n)$. Use the standard topology on $\mathcal{S}$ $ \|f\|_{a,b}=\sup_{x \in \Bbb{R}^n}\left| x^a\partial^bf \right| $ with $a,b\in\Bbb{Z}_+^n$.
This post highligths the proof: Let $g \in \mathcal{S}$, there is a sequence $\{f_n\} \subset \mathcal{D}$ for which $ f_n\ast g \to g \quad \text{in} \quad L^1(\Bbb R^n). $ Hence, there is a subsequence $\{h_n\}$ of $\{f_n\}$ for which $ h_n \ast g \to g \quad \text{a.e.} $ How to show $ \|h_n\ast g -g \|_{a,b} \to 0 \quad \text{for any} \quad a,b? $