Let $\Omega$ be an open bounded set in $R^n$ with smooth boundary. Suppose that $u_n$ is a sequence in $H^1_0(\Omega)$ which converges weakly to $u$ in the sense that for all $y \in H^1_0(\Omega)$
$$\int_\Omega \nabla u_n\cdot \nabla y \to \int_\Omega \nabla u\cdot \nabla y$$
Then, why is it true that $u_n$ converges to $u$ weakly in $L^2(\Omega)$, that is,
$$\int_\Omega u_n\cdot y \to \int_\Omega u\cdot y$$ for all $y \in L^2(\Omega)$?