This is more or less a question about a technical detail in Evan's PDEs Chapter 6 problem 4. Let $\Omega$ be an open, connected, bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Consider the subspace $H=\{u\in H^1(\Omega): \int_{\Omega} u\,\text{d}x=0\}$ with the inner product $\langle u,v\rangle_H:=\int_{\Omega}\nabla u\cdot \nabla v\,\text{d}x$. It is easy to see that this is indeed an inner product (relying heavily on the connectedness hypothesis). However I would like to go further and see that it is a Hilbert space.
I start with a Cauchy sequence $\{u_k\}_{k=1}^\infty$ in $H$, which by definition means that $\int_{\Omega}\lvert \nabla(u_m-u_n)\rvert^2\,\text{d}x\to 0$ as $m,n\to\infty$. Now I want to demonstrate that this sequence has a limit and I feel like I should be using Fatou's lemma or something, but I have gotten nowhere and I'm trying to interchange these lim infs and (weak) gradients and it seems super sketchy. Any help would be amazing.