I'm trying to understand a proof that my professor wrote in his notes, which is the following.
$W^{k,p}(\Omega)$, with $\Omega$ an open subset of $\mathbb{R}^n$, is a complete Banach space.
He considered the specific case of $H^1(\Omega)$, but stated that it would be the same procedure for the general case and I understand that.
To prove the completeness, he, of course, started by introducing a Cauchy sequence ${u_k} \subset H^1(\Omega)$, which would imply that $$\left\lVert u_h - u_k \right\rVert _{L^2} \to 0\hspace{1mm} ; \hspace{6mm} \left\lVert \partial_j u_h - \partial _j u_k \right\rVert_{L^2} \to 0 , \ j= 1, \ldots,n$$ Here I don't understand why he assumes that ${\partial_j u_k}$ is also a Cauchy sequence just by saying that ${u_k}$ is a Cauchy sequence.