Proof that $C_{0} ^{\infty}$ bounded functions are dense in $L^{1} (\mathbb{R})$

Question

We have been told that functions $C^{\infty}$,and with a compact support are dense in $L^{1} (\mathbb{R})$. While I can prove them singularly (except for the functions $C^{\infty}$), I cannot find, given $f \in L^{1}$ a succession $\{f_n \}$ that satisfies both of them. Also, I was wondering if these kind of subspaces (as well as other dense subspaces like step functions for instance) are also dense in every $L^{p}$, and (if not) if there is a standard method for checking out if a particular dense subspace of $L^{1}$ is dense for other $L^{p}$. Thanks for the help!