The restriction of functions in $H^1(\mathbb{R}^n)$ to a bouded open set $\Omega$ are in $H^1_0(\Omega)$?

Question

I'm new in studying PDEs and in the book I'm reading, a lot of times the author make the assertion: since $\|u_n\|_1$ is a bounded sequence in $H^1(\mathbb{R}^n)$ then we can assume, taking a subsequence if necessary, that there exists $u\in H^1(\mathbb{R}^n)$ such that $u_n$ converges weakly to $u$ and $u_n\to u$ in $L_{loc}^{p}(\mathbb{R}^n)$. The first assertion is ok for me, but the second one is weird. My thinking is that if we take an open set $\Omega\subset\mathbb{R}^n$ such that its closure is compact, then the restriction of the $u_n$ to such a set is in $H^1_0(\Omega)$, so there exists a compact embedding of $H_0^1(\Omega)$ in $L^p(\Omega)$ (apropriated $p$) so the assertion will be true if this was the case. But is this argument valid? The restriction of a function in $H^1(\mathbb{R}^n)$ to a bounded open set $\Omega$ is in $H_0^1(\Omega)$?, i.e. if $v\in H^1(\mathbb{R}^n)$ implies that $v_{|\Omega}\in H_0^1(\Omega)$?