How to rigourously prove that any discrete functions could be interpolated? Define a discrete function $g:\mathbb N\to\mathbb R, \ n\mapsto g(n)$
How to prove that the values of $g(n)$ could be interpolated by a smooth function:
$$\exists f\in\mathcal C^\infty:\mathbb R\to\mathbb R$$
s.t. $f(n)=g(n)$.
This seems obvious but I cannot find a elementary way to do this. I had tried two things:
1) Suppose $f(x)\in\mathcal C^\omega$, expend $f(x)=\sum_{m\in\mathbb N} a_mx^m$, substitute in $(n,g(n))$ and solve for $a_m$. There are two downsides. First I am not sure if the Taylor series converge in the whole domain. Second is I am not sure if it correct to solve an infinite dimensional vector $a$ like this.
2) Zero dimensional manifold can always be immersed in a first dimensional smooth manifold. 

I see. The original question was just about $C^\infty,$ I was thrown off by the mention of $C^\omega;$ this answer gives  $C^\omega.$
Solve the exercise in Ahlfors for $f(z).$ As your $n$ and $g(n)$ are real, this means that the real part of $f(z)$ is a harmonic function on the whole $x,y$ plane that solves your problem; it is real analytic. Therefore its restriction to the real line is also real analytic. 

