Let $\Omega$ be an open bounded of $\mathbb{R}^n$. I know that $u$ is Lipschitz continuous if and only if $u\in W^{1, \infty}(\Omega)$. Now, if I consider the norm $$ I_{\infty}:W^{1, \infty}(\Omega)\longrightarrow\mathbb{R},\qquad I_{\infty}(u, \Omega):=\|Du\|_{L^{\infty}(\Omega)} $$ where $\|Du\|_{L^{\infty}(\Omega)}=\operatorname{ess\ sup}_{\Omega}|Du|$, with $|Du|=\max_{i=1,\ldots,n}|u_{x_i}|$, I do not understand why we can conclude that if $u$ is Lipschitz continuous (and then it belongs to $W^{1, \infty}(\Omega)$), then $I_{\infty}(u, \Omega)<+\infty$.
Some helps?
Thank You