Let $U$ be an open bounded set in $\mathbb{R}^n$ and let $u_i\in W_0^{k+2,p}(U)$ with $W_0^{k+2,p}(U)$ denoting the closure of $C_c^\infty(U)$ in the Sobolev space $W^{k+2,p}(U)$. I'm trying to show that if $\Delta u_i=f_i$, and $f_i$ is a Cauchy sequence in $W^{k,p}$, then $u_i$ is a Cauchy sequence in $W^{k+2,p}$. Here's my strategy:
By applying the Rellich-Kondrachov theorem, I was able to find a constant $C>0$ s.t. $\forall u\in W_0^{k+2,p}$, $$\lVert u\rVert_{W^{k+2,p}}\ \leq C\lVert\Delta u\rVert_{W^{k,p}}\ .$$ Fix $\epsilon>0$. Since $f_i$ is Cauchy in $W^{k,p}$, $\exists N\in\mathbb{N}$ s.t. $i,j\geq N\Rightarrow\lVert f_i-f_j\rVert_{W^{k,p}}<\frac{\epsilon}{C}$. Now, if $i,j\geq N$, then $$\lVert u_i-u_j\rVert_{W^{k+2,p}}\ \leq C\lVert f_i-f_j\rVert_{W^{k,p}}<\epsilon.$$ This shows $u_i$ is Cauchy in $W^{k+2,p}$.
Did I miss anything? Thank you.