For $\epsilon>0 $ define $g_\epsilon(u)=\sqrt{\epsilon^2+u^2}-\epsilon$.
One finds $ \nabla g_\epsilon(u)=\frac{u}{\sqrt{\epsilon^2+u^2}}\nabla u$ and $ g_\epsilon(u)\in W_0^{1,2}(\Omega)$ .
Then it holds $\int_{\Omega}|\nabla g_\epsilon(u)|^2dx \leq \int_{\Omega}|\nabla u|^2dx$
For some sequence ${n_k} \subset \mathbb{N}$ it follows
$\int_{\Omega}|\nabla |u||^2dx=\int_{\Omega}\lim\limits_{k \rightarrow \infty}|\nabla g_\frac{1}{n_k}(u)|^2 dx=\lim\limits_{k \rightarrow \infty}\int_{\Omega}|\nabla g_\frac{1}{n_k}(u)|^2 dx \leq \int_{\Omega}|\nabla u|^2dx $ (1)
I do not really understand what (1) says .