Version of Stone Weierstrass for functions not vanishing at infinity I am trying to see what is known about uniform density of function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$ (bounded continuous functions on $\mathbb{R}^n$). By uniform density, I mean density with respect to uniform convergence on the whole of $\mathbb{R}^n$, not any compactly supported version of uniform convergence. I do not know of any version of Stone Weierstrass theorem covering such cases. But I would be happy to see certain special examples where people have proved density of certain function spaces in $C(\mathbb{R}^n)$ or $C_b(\mathbb{R}^n)$. I am just curious about this, I do not have an especially good reason for asking this. As a possible example, if someone has proved somewhere that continuous functions of bounded smooth functions are dense in $C(\mathbb{R}^n)$, I would be interested in learning how such a proof might work. Thanks!
Thanks!

Even though the sup-norm is not a finitely-valued function on $C(R^n)$ we can can still talk of uniform convergence, and density. However even with $n=1$, if $\{f_k : k \in N \}$ is a subset of $C(R)$, there exists $g \in C(R)$ with $g(k)=1+f_k(k)$ for each $k \in N$.
