Approximating Sobolev functions on unbounded domains of class $C^1$ by compactly supported smooth functions

In the lecture we saw the following statement

Corollary: Let $\Omega$ be open and of class $C^1$, $1 \leq p < \infty$ and $u \in W^{1,p}(\Omega)$. Then there exists a sequence $(u_k)_{k \in \mathbb{N}} \subset C^{\infty}_c(\mathbb{R}^n)$ with $||u_k|_{\Omega}-u||_{W^{1,p}(\Omega)} \rightarrow 0, k \rightarrow \infty$

The proof went as follows:

First we set $\delta>0$ and constructed a $\overline{u} \in W^{1,p}(\Omega)$ with $supp(\overline{u}) \subset\subset \mathbb{R}^n$, i.e. compact support in $\mathbb{R}^n$, with $||\overline{u}-u||_{W^{1,p}(\Omega)} < \delta$. Then the proof said to use a extension operator for Sobolev functions, which can be constructed, since for the boundary $\Gamma$ of $\Omega$ it holds that $\Gamma \cap supp(\overline{u})$ is compact. This was the part I did not understand. For reference our lecturer used the following notes. The above stated corollary is Korollar 8.4.2. Unfortunatly the notes are written in german, but since the notes are mainly based on Brezi's book on functional analysis, the same statement can be found in Corollary 9.8 in Brezi's book.

Thanks a lot in advance!