Cauchy sequence in $W^{1,2}$ is Cauchy in $L^2$?

Question

I'm trying to show that the space $W^{1,2}$ is an Hilbert space, i found this answered question: Showing Sobolev space $W^{1,2}$ is a Hilbert space

which offer a proof, but I'm having trouble with one of the basic steps: in order to claim that if $f_n$ is Cauchy in $W^{1,2}$ than $f_n \rightarrow f$ in $L^2$ , I need to show that $f_n$ is Cauchy in $L^2$ , correct? How can I show that?

Thanks Shahar

Answer 1

Since $\lVert f\rVert_{1,2}\ge \lVert f\rVert_2$, $$\limsup_{\min(n,m)\to \infty}\lVert f_n-f_m\rVert_{2}\le\limsup_{\min(n,m)\to \infty}\lVert f_n-f_m\rVert_{1,2}=0$$