This might be a known result, but I haven't found anything so far.
Prove or disprove:
For any real sequence $\left(u_n\right)_{n\in\mathbb{N}}$, there is at least one analytic function $f$ such that $f$ coincides with $\left(u_n\right)_{n\in\mathbb{N}}$ on $\mathbb{N}$, i.e $\forall n\in\mathbb{N}, f(n)=u_n$.
I haven't any clue as of how to proceed...
If this result happens to be false, then what about this weaker version :
For any real sequence $\left(u_n\right)_{n\in\mathbb{N}}$, there is at least one smooth function $f$ such that $f$ coincides with $\left(u_n\right)_{n\in\mathbb{N}}$ on $\mathbb{N}$, i.e $\forall n\in\mathbb{N}, f(n)=u_n$.
I have no idea wether the first conjecture is true or false, but I am pretty sure the weaker one is true. For proving the weaker conjecture, I have thought about building a smooth function by doing "smooth junctions" between pieces of linear functions that would link the values taken by $\left(u_n\right)_{n\in\mathbb{N}}$. Seems doable, albeit pretty messy. But for the stronger conjecture, I don't think "analytic junctions" are always possible...