I have taken $u \in W^{1,p}(\mathbb{R}^n)$ and a countable cover of $\mathbb{R}^n$ by open balls of increasing radius. I was hoping to mollify and use a partition of unity to deduce that $C^{\infty}_c(\mathbb{R}^n)$ is dense in $W^{1,p}(\mathbb{R}^n)$ but this method does not give approximation by functions of compact support as far as I can see.
What is the best way to deduce this density?