Consequence of a Sobolev's embedding into $L^{\infty}$

Question

Hy everyone. I would to like to take a question about the following Sobolev's inequality - for $p>n$, $\Omega \subset \mathbb{R}^{n}$ bounded open set with $C^{1}$ boundary

$\| u\|_{L^{\infty}(\Omega)} \leq \| u\|_{W^{1,p}(\Omega)}$

A consequence of this is that the function $u$ is continuous? In this topic topic the most voted answer says yes for this question, because the smooth elements of $W^{1,p}(\Omega)$ are dense therein, but i do not understand the link between my question, the density argument and the inequality fixed in this topic. Any tips?

Answer 1

If $(u_k)\subset C^\infty(\overline{\Omega})$ converges to $u \in W^{1,p}(\Omega)$, then the above inequality applied to $u_k-u_l$, shows that $(u_k)$ is Cauchy in $C(\overline{\Omega})$ (with the standard $\sup$ norm). This means that the limiting function $u$ has an a.e. representative that is continuous in $\Omega$.