Question: Let $\{u_i\}$ and $\{f_i\}$ be two sequences of functions in $C^\infty$, and suppose that $u_i\to u$ in the Sobolev space $W^{k+2,p}$ and that $f_i\to f$ in $W^{k,p}$. If each $u_i$ solves the Poisson equation $\Delta u_i=f_i$ strongly(omitted hereafter since we won't consider weak solutions), can we conclude that $u$ solves $\Delta u=f$?
Attempt: Certainly I could do something if uniform convergence were assumed. But all I have is convergence in the Sobolev spaces. So I turned to $$\lVert\Delta u-f\rVert_{W^{k,p}}=0$$ for help. Now the triangle inequality gives $$\lVert\Delta u-f\rVert_{W^{k,p}}\leq\underbrace{\lVert\Delta u-f_i\rVert_{W^{k,p}}}_{(1)}+\underbrace{\lVert f_i-f\rVert_{W^{k,p}}}_{(2)}\ .$$ The second term (2) on the LHS surely converges to zero by hypothesis, and the first term (1) can be written as $$\lVert\Delta(u-u_i)\rVert_{W^{k,p}}\ .\tag{3}$$ I was trying to use convergence of $\{u_i\}$ to vanquish (3), but nothing useful has ever come into my mind. Is there any problem in my attempt? Thank you.