Let $1<p<\infty$ and the the sequence $\{u_n\}$ is bounded in the Sobolev space $W^{1,p}_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$. Then by compact embedding we have $u_n\rightarrow u$ in $L^p(\Omega)$ (upto a subsequence). Also we have $\{|\nabla u_n|\}_n$ is bounded in $L^p(\Omega)$. So it implies that upto a subsequence $\nabla u_n$ is weakly convergent.
Now the question that is it true that upto a subsequence $\nabla u_n\rightarrow \nabla u$ alomost everywhere?.