Let $\Omega = (0, 1) \times (0, 1) \subset \mathbb{R^2}$ and let $f :\Omega \to \mathbb{R}$ be a Lipschitz continuous function. Prove that $f \in W^{1,\infty}(\Omega)$.
My attempt:
let $\epsilon>0$ , $\Omega_\epsilon = \{x\in \Omega : B_\epsilon(x)\subset \Omega\}$, and $f_\epsilon= J_\epsilon\ast f$ be the mollification. Then $f_\epsilon \in C^\infty(\Omega_\epsilon)$. We can approximate;
\begin{align} \|f_\epsilon\|_{ W^{1,\infty}(\Omega_\epsilon)} \le \|D(J_\epsilon\ast f)\|_{ L^{\infty}(\Omega_\epsilon)} \le \|Df\|_{ L^{\infty}(\Omega_\epsilon)} \le \|f\|_{C^1(\Omega_\epsilon)}< \infty \end{align}
letting $\epsilon \to 0 $, we have that $f_\epsilon \to f$ unifromly in $ W^{1,\infty}(\Omega)$.
