Let $H = H_{0}^{1}(\Omega)$ where $\Omega$ is a bounded domain in $R^N$ whose boundary $\partial\Omega$ is a smooth manifold. We know that the embedding $$H\hookrightarrow L^s(\Omega)$$ is compact for each $s\in [1, 2^*)$ and continuous for each $s\in [1, 2^*]$
Suppose that $\{u_n\}$ is a sequence in $H$ such that $u_n\rightharpoonup u$. Then $\{u_n\}$ is bounded, so by the compact embedding there exists a subsequence $\{u_{n_k}\}$ such that $u_{n_k}\rightarrow u_0$ in $L^s(\Omega)$ for $s\in [1, 2^*)$
How do we know that $u_0 = u$?