Continuous Functions with Compact Support Dense in Lebesgue P-Space

Theorem
Let $C_c \left({\R^n}\right)$ be the space of continuous functions with compact support on $\R^n$.

Let $p \in \R$, $p \ge 1$, and let $\mathcal L^p \left({\lambda^n}\right)$ be Lebesgue $p$-space for Lebesgue measure $\lambda^n$.

Then $C_c \left({\R^n}\right)$ is everywhere dense in $\mathcal L^p \left({\lambda^n}\right)$ with respect to the $p$-seminorm $\left\Vert{\, \cdot \,}\right\Vert_p$.