Denote by $W_0^{k, q}$ the closure of $C_0^{\infty}\left(\mathbb{R}^n\right)$ in $W^{k,q}(\Omega)$ and by $W_c^{k, q}(\Omega)$ the set of functions from $W^{k, q}(\Omega)$ that have compact support in $\Omega$. I was able to show that $$ W_c^{k, 2}(\Omega) \subset W_0^{k, 2}(\Omega) $$ and now I wanted to ask if there is a proof for $W_c^{1, q}(\Omega) \subset W_0^{1, q}(\Omega)$.
I have looked in some books (Sobolev Spaces by Adams, PDE by Evans) but without success.