[Math] The subset of $C^{\infty}$ functions with compact support in $\mathbb{R}$ in the space of bounded real valued continuos function on $\mathbb{R}$

Question

could any one tell me the following statement is true or false? and any reference for proof or counter examples?

The subset of $C^{\infty}$ functions with compact support in $\mathbb{R}$ in the space of bounded real valued continuous function on $\mathbb{R}$ is dense

Answer 1

Obviously false. There is no sequence in $C_c(\mathbb R)$ (of smooth or "rough" functions) that converges uniformly to the constant function $1$.

Indeed if $(f_n)$ is any sequence in $C_c(\mathbb R)$, then $\displaystyle \lim_{n\to\infty} \lVert 1-f_n\rVert_\infty \geq 1$ since each $f_n$ is compactly supported.