Does Weak convergence in $W^{1,p}(\Omega)$ implies almost everywhere of the gradient $\nabla$

Question

Let $\Omega$ be an open bounded subset of $\mathbb{R}^N$ and $1<p<N$. Let $\{u_n\}$ in $W_0^{1, p}(\Omega)$ be such that $$ u_n \rightharpoonup u \quad \text { in } W_0^{1, p}(\Omega) . $$

Then, it is true that up to subsequence $$ \nabla u_n \rightarrow \nabla u, $$ almost everywhere in $\Omega$ It was on my notes, so I guess it is true.

Could anyone help, please?

Thank you in advance.