# Can a holomorphic function satisfy f(1/n)=1/(n+1)?

Does there exist a function $f$ which is holomorphic on $B_0(2)$ (open disc of radius 2 in the complex plane) such that $f(1/n)=1/(n+1) \forall n \in \mathbb{N}$? At the moment I'm thinking not but a proof is seeming elusive. Any hints would be appreciated.

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Let $a_n:=\frac 1n$. Such a function should satisfy $f(a_n)=\frac{a_n}{1+a_n}$. Let $g(z):=f(z)-\frac z{z+1}$ on $B(0,1)$, the open ball. What can you say about the zeros of $g$?