Can any function on naturals be interpolated to a smooth function on reals? Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be an arbitrary function from naturals to naturals. Is it always possible to find a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that

*

*for any $n \in \mathbb{N}$, we have $f(n) = g(n)$, and

*$g \in C^\infty$?

I'm asking because I was trying to prove a result about the ratios of functions from naturals to naturals and it occurred to me that if I could always interpolate to get back smooth functions from integers to integers, I could conceivably use l'Hopital's rule to resolve the limits.
Thanks!

Given a sequence of points $z_n$ in an open subset $\Omega$ of $\mathbb{C}$ without an accumulation point, and $\alpha_n$ a sequence of values, there exists a holomorphic function $\phi$ on $\Omega$ with $\phi(z_n)= \alpha_n$, for all $n$. Now, if $z_n$ and $\alpha_n$ are real, we can take $\phi$ real on $\Omega\cap \mathbb{R}$ ( take
$\frac{\phi(z) + \bar \phi(\bar z)}{2}$).
This is a standard result in complex analysis.
