Let $\Omega\subset\mathbb R^n$ be open and bounded. I consider a sequence $u_k:\Omega\to\mathbb R$ of smooth functions which converges uniformly to a function $u:\Omega\to\mathbb R$. Moreover, the sequence of gradients $\nabla u_k$ is uniformly bounded. Now I would like to show that $$ \int_\Omega |\nabla u|~dx\leqslant \liminf_{n\to \infty}\int_{\Omega}|\nabla u_k|~dx. $$ References and counterexamples are very welcome.
EDIT: The integral on the lhs exists since the sequence is uniformly Lipschitz and $u$ has therefore (since it is Lipschitz) a gradient defined almost everywhere.