I have the following statement:
Let $K\subset \mathbb{R}$ be a compact set and define $H^s_K(\mathbb{R})=\{u\in H^s(\mathbb{R}); supp\,\,u\subset K\}$, for $s\in \mathbb{R}$. Then $H^s_K(\mathbb{R})$ is a closed subset of $H^s(\mathbb{R})$.
Here, $supp\,\, u$ denotes the support of $u$ and $H^s$ the Sobolev space of functions in the Schwartz space.
To prove it, I have tried to take a sequence $(f_n)\subset H^s_K$ converging to $f\in H^s$, but I cannot conclude that $supp\,\,f\subset K$, although it makes sense to think that considering that $supp\,\, f_n \subset K$ for all $n$.
Any ideas on the proof?