Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, let $C^{0,1}_0(\Omega)$ be the set of all Lipschitz continuous functions on $\Omega$ that vanish on the $\partial \Omega$. Let $C^{\infty}_c(\Omega)$ be the set of all infinitely differentiable functions with compact support on $\Omega$.
I would like to prove that $C^{\infty}_c(\Omega)$ is dense in $C^{0,1}_0(\Omega)$ in the standard Holder space norm \begin{equation*} ||u||_{C^{0,1}(\Omega)} := \sup_{x \in \Omega} |u(x)| + \sup_{x,y \in \Omega; x \neq y} \frac{|u(x)-u(y)|}{|x-y|}. \end{equation*}