$L^{2}(\mathbb{R})$ is a separable Hilbert space.

Question

I want to show $L^{2}(\mathbb{R})$ is separable. My idea is $C_{c}(\mathbb{R})$ is dense in $L^{2}(\mathbb{R})$ in $L^2$ norm and polynomials with rational coefficients are dense in $C[a,b]$ in $\sup$ norm and hence in $L^2$ norm. Is my idea correct?