If $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$ with $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ then $u_k \to u $ in $W^{1,p}(\bar{\Omega})$

Question

I have already proven that for every $u \in W^{1,p}(\Omega)$ with $1 \leq p < \infty$ and every open set of class $C^1$ there exists a sequence $(u_k) \subset C_c^{\infty}(\mathbb{R}^n)$ such that $u_{k|_{\Omega}} \to u$ in $W^{1,p}(\Omega)$.

I am now curious if this implies that $u_k \in C^{\infty}(\overline{\Omega}) $ and converges to $u$ in $W^{1,p}(\Omega)$, due to the fact that $\Omega $ is $C^1$.