I'm trying to prove that $C_c^{k}(\mathbb{R}^n)=C_c(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$.
I know that $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$ so $C_c(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$.
$C^{k}(\mathbb{R}^n)$ is trivially dense in $L^p(\mathbb{R}^n) \cap C^{k}(\mathbb{R}^n)$.
Using that result Intersection of Dense Sets, I see that it is sufficient that one of the dense sets is open. How can I prove it?