The following theorem is presented as question 15.6 in Treves (1967, 1995). $\mathscr{S}$ is the Schwartz space of functions on $\mathbb{R}^n$, and when one says functions on $\mathbb{C}^n$ are 'dense' in the space, one naturally has in mind their restriction to the reals.
Theorem. $\hspace{0.05cm}$ The entire analytic functions which belong to $\mathscr{S}$ are dense in $\mathscr{S}$.
If we use the fact that $C_c^{\infty}(\mathbb{R}^n)$, the set of "bump" functions, is dense in $\mathscr{S}$, it remains to prove that the analytic functions in $\mathscr{S}$ are dense in $C_c^{\infty}(\mathbb{R}^n)$ in the Schwartz topology.
My approach has been to design a sequence of entire analytic functions $\{f_n\}$. A bump function $f$ is the limit, in $C^{\infty}(\mathbb{R}^n)$, of a sequence of polynomials $\{p_n\}$. Let $f_n=e^{-|x|^2}p_n(x)$. Then $f_n$ belongs to $\mathscr{S}$ and can be extended as an analytic function.
It remains to show that:
(a) $f_n$ converges to $f$ in $C^{\infty}(\mathbb{R}^n)$
(b) $f_n$ is Cauchy in $\mathscr{S}$.
I am unable to show (b), nor have an a priori reason to believe it should be true. My sense is that I've been working too hard, and taking a bit of a chance, in how I've approached this. I was wondering if someone could see a simpler approach.