Completeness of sobolev functions

Question

I am looking for a proof of the statement that the completion of $C^{\infty}$ functions with compact support in the $L^{\infty}$ norm is the space of continuous function which vanishes at infinity. Can some body help?

Answer 1

Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function which vanishes at infinity and fix $\epsilon>0$. Then there exists a compact set $K$ such that $|f|<\epsilon/2$ outside of $K$ and another compact set $K' \subset K$ such that $|f|<\epsilon$ outside of $K'$. On the larger set $K$, apply the Stone-Weierstrass theorem (p. 13 for $\mathbb{R}^n$ version) to get a polynomial $g$ satisfying $$|f(x)-g(x)|<\epsilon/2 \qquad x \in K.$$ By the "smooth Urysohn's lemma" there exists a smooth positive function $0 \leq h \leq 1$ with support in $K$ so that $h(x) \equiv 1$ for $x \in K'$. The function $gh$ is smooth with support $K$; simply check that $|f-gh|$ is small everywhere.