I consider $u\in C^{0,1}(\Omega)$ a Lipschitz function with $\Omega\subset\mathbb{R}^n$ open.
Let be $[u]_{C^{0,1}(\Omega)}=\sup _{x,y\in\Omega, x\neq y}\dfrac{|u(x)-u(y)|}{|x-y|}$ the Lipschitz constant of $u$.
I consider $u_\epsilon=u*\rho_\epsilon$, defined on $\Omega_\epsilon=\{x\in \Omega |dist(x,\Omega)>\epsilon\}$,
where $\rho_\epsilon(x)=\epsilon^{-n}\rho(\dfrac{x}{\epsilon})$ and $$\rho(x) = \begin{cases} e^{1/(1-|x|^2)}/I_n& \text{ if } |x| < 1\\ 0& \text{ if } |x|\geq 1 \end{cases},$$ ( $I_n$ is a coefficient of normalization).
How can I prove that $[u_\epsilon]_{C^{0,1}(\Omega_\epsilon)}\leq[u]_{C^{0,1}(\Omega)}$?
I don't know how to do it. Can someone help or give me a reference? Thank you!