Let $\Omega \subset \mathbb{R}$ be a bounded open set with $C%1$ boundary and $H^1(\Omega) = W^{1,2}(\Omega)$ be the Hilbert-Sobolev space. Let ${u_n}$ be a sequence of functions which are uniformly bounded in $H^1(\Omega)$. If I know that $\|u - u_n\|_{L^2(\Omega)} \rightarrow 0$ how do I show that $u \in H^1(\Omega)$ and that \begin{equation} \|u\|_{H^1} \leq \liminf_{n \rightarrow \infty} \|u_n\|_{H^1} \end{equation} I don't know much about Sobolev spaces apart from some readings on the Embedding Theorems so I don't really know how to begin.
So is the main point of this problem is to show that $\|D_{x_i}u\|_{L^2(\Omega)} < \infty$ for all coordinate variables $x_i$ (so that $\|u\|_{H^1} < \infty$ hence $u \in H^1(\Omega)$)? I don't see why the inequality I'm required to show should be true either.
Please guide me.
The following outlines the necessary steps, but some details are left out:
First choose a subsequence $(u_{n_k})_k$ with $\Vert u_{n_k} \Vert_{H^1} \to \liminf_n \Vert u_n \Vert_{H^1}$ and rename the sequence to $(u_n)_n$ again.
You know that every bounded sequence has a weakly convergent subsequence (because $H^1$ is a Hilbert space).
Hence $u_{n_k} \to v$ for a suitable subsequence and some $v \in H^1$, where the convergence is weak convergence in $H^1$. By weak lower continuity of the $H^1$-norm, we get $\Vert v \Vert_{H^1} \leq \liminf_k \Vert u_{n_k} \Vert_{H^1} = \liminf_n \Vert u_n \Vert_{H^1}$.
Finally, using the continuous embedding $H^1 \hookrightarrow L^2$, we also get $u_{n_k} \to v$ weakly in $L^2$. But also $u_{n_k} \to u$ strongly in $L^2$, hence $v=u$, which easily completes the proof.