Let $\Omega\subset\mathbb{R}^3$ be a bounded simply-connected domain. Consider the following subspace of $L^2(\Omega)$ $$ G(\Omega) = \{\mathbf{u}\in L^2(\Omega)\colon\mathbf{u}=\nabla p\textrm{ for some }p\in H^1(\Omega)\}, $$ where $H^1(\Omega)$ is the Sobolev space $W^{1,2}(\Omega)$ and $\nabla p$ is understood as the weak derivative. Show that $G(\Omega)$ is closed in $L^2(\Omega)$. I very much prefer a proof without using mollifiers if possible!
The naive approach got me nowhere: consider any sequence $(u_n) = \nabla p_n\longrightarrow u$ in $L^2(\Omega)$, I tried to identify $u$ as $\nabla p$, where $p$ is the limit of $p_n$ in $L^2(\Omega)$, but this sounds wrong to me.