$C(\mathbb{R})\cap L^p(\mathbb{R})$ is dense in $L^p(\mathbb{R})$

Question

How prove that, $C(\mathbb{R})\cap L^p(\mathbb{R})$ is dense in $L^p(\mathbb{R})$ ? Thanks

Answer 1

We have that $C_c(\mathbb{R})$ are dense in $L^p(\mathbb{R})$ and $C_c(\mathbb{R})\subset C(\mathbb{R})∩L^p(\mathbb{R}) \subset L^p(\mathbb{R})$.