Let $\Omega$ be a bounded open with no assumptions on the boundary $\partial\Omega$. Show that $H^{k+1}_0(\Omega)$ is compactly embedded in $W^{k,2}(\Omega)$.
Show that if $\Omega$ is bounded, then $L^2(\Omega)$ is compactly embedded in $H^{−1}(\Omega)$
By Wikipedia,(https://en.wikipedia.org/wiki/Compact_embedding) I know I have to satisfy two conditions. I'd appreciate a full, clear answer on how to do these two exercises.