Compactly supported functions are dense in local Sobolev spaces

Question

Let $\Omega\subset\mathbb{R}^n$ be an open set. I would like to know why $\mathcal{D}(\Omega,\mathbb{C}^l)=C^{\infty}_{\mathrm{c}}(\Omega,\mathbb{C}^l)$ is dense in $H^s_{\mathrm{loc}}(\Omega,\mathbb{C}^l)$ for all $s\in\mathbb{R}$. Is it true that it is also dense in $H^s_{\mathrm{c}}(\Omega,\mathbb{C}^l)$?