Smooth Bump Functions are Square Integrable

Question

I am currently trying to prove that the smooth functions with compact support on $R^{n}$ (i.e. smooth bump functions) $C^{\infty}_{0}(\mathbb{R}^n)$ are a subspace of $L^{2}(\mathbb{R^n})$, i.e., the space of square integrable functions on $\mathbb{R}^n$.
I'm a little rusty as I haven't proven anything involving integrals for a while; I know it's probably a pretty straightforward proof. I'm not sure if it's simply enough that $f\in C^{\infty}_{0}(\mathbb{R}^n)$ is continuous and bounded on $\mathbb{R}^n$, or if more needs to be shown. And while it is true that $f$ is continuous and bounded, I'm not sure if it follows that $|{f}|^2$ is continuous.
Thank you for any help!

Answer 1

Yes $|f|^2$ is continuous, since compositions of continuous functions are continuous. Since it vanishes off a bounded set and is bounded, it's integrable.