Showing $\|u\|_{L^{\infty}}\le C\|u\|_{W^{1,p}}$.

Question

I was reading the proof from Brezis' book that show that if $p>N$ we have $W^{1,p}(\mathbb{R}^N)\subset L^{\infty}(\mathbb{R}^N)$. He starts proving $\|u\|_{L^{\infty}}\le C\|u\|_{W^{1,p}}$ for fonctions $u\in C^{\infty}_c(\mathbb{R}^N)$ and than concludes using density. I was wondering how exactly to do this. I think $C^1_c(\mathbb{R}^N)$ is dense in $W^{1,p}$, so if we have $u\in W^{1,p}$ we can consider a sequence $u_n\to u$ where the $u_n$ are in $C^1_c(\mathbb{R}^N)$. And we have $\|u_n\|_{L^{\infty}}\le C\|u_n\|_{W^{1,p}}$. But then how to conclude for $\|u\|_{L^{\infty}}\le C\|u\|_{W^{1,p}}$? What result is used here?