Denseness of bounded funtions in L^2?

[logarithmic]

Let $C_b^\infty(\mathbb{R}^n)$ be the space of infinitely differentiable functions f, such that f and all its partial derivatives are bounded.

Is $C_b^\infty(\mathbb{R}^n)$ dense in $L^2(\mathbb{R}^n)$? I think the answer is yes, because $C_b^\infty(\mathbb{R}^n)$ contains $C_0^\infty(\mathbb{R}^n)$, the space of all infinitely differentiable functions with compact support, as a subset. And it's well known that $C_0^\infty(\mathbb{R}^n)$ is dense in $L^p(\mathbb{R}^n)$.

However, there appear to be functions in $C_b^\infty(\mathbb{R}^n)$ but are not in $L^2(\mathbb{R}^n)$, for example the function f(x)=1. So this means that instead, we have $C_b^\infty(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)$ dense in $L^2(\mathbb{R}^n)$?

 

[mathman]

You seem to have answered your own question.