I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$.
I know that I can't directly use the compactness of Rellich Kondrachov Theorem since I am taking $p = \infty$. From Morrey's Inequality I have $||u||_{C^{0,\alpha}} \leq ||u||_{W^{1,p}}$ where $\alpha = 1 - \frac{n}{p}$. Is it possible to take this further and show that $W^{1,p}(\Omega) \Subset L^{\infty}(\Omega)$ where $\Omega \subset \mathbb{R}^{n}$ is $C^{1}$?