Is $H_0^1(\mathbb{R}) \cap H^2(\mathbb{R})$ compactly embedded into $L^2(\mathbb{R})$?

Question

I am trying to prove that $H_0^1(\mathbb{R}) \cap H^2(\mathbb{R})$ compactly embedded into $L^2(\mathbb{R})$. I have been using the template of the proof from the Rellich-Kondrachov theorem but I cannot find precisely what I want... Does anyone have a nice way to prove this result?

Best wishes,

Catherine