Let $g: \mathbb{R}^n \to \mathbb{R}$ be a continuous function, such that $g(x)>0$, for all $ x \in \mathbb{R}^n$. I want to prove that: there is an entire analytic function $h$ (this means that $h$ can be extended to $\mathbb{C}^n$ as a function everywhere holomorphic) such that, $$0 < h(x) < g(x),\; \forall \; x \in \mathbb{R}^n .$$
If $g$ has compact support, then, I know there exists a sequence $(f_k)_{k \in \mathbb{N}}$ of entires functions in $\mathbb{R}^n$ so that, for each, $k \in \mathbb{N}$, the function $f_k$ can be extended to the complex values as an entire function and $f_k \to g$ uniformly in $\mathbb{R}^n$ .
But, $g$ has compact support?
The only idea I had was involving the support of $ g $.
From there, how do I proceed? This is the way?