# Closure of smooth compactly supported functions separable

Consider the norm $$\|f\|=\sup_{x>0}|xf^\prime(x)|$$ on the space $$\mathcal{C}_c^\infty(0,\infty)$$, the space of smooth functions with compact support in $$(0,\infty)$$.

I want to prove that the closure of $$\mathcal{C}_c^\infty(0,\infty)$$ in the norm $$\|\cdot\|$$ is separable.

Every function in $$C_c^\infty(0;\infty)$$ has support contained in some $$[1/n;n]$$ where $$n\in \mathbb{N}$$. However, on $$[1/n;n]$$ we can approximate using polynomial functions (by the Stone-Weierstrass theorem). To render it countable, we can use polynomial function with rational coefficients. Now we just need to extend the polynomial functions to all of $$(0;\infty)$$. This we can do by multiplying with a cut-off function with suitable support. Hence, the space is separable and therefore also its closure.