If $\{z_n\}$ is a sequence of distinct complex numbers in unit circle such that $z_n \rightarrow 0$ as $n \rightarrow \infty$, there exist an entire function $f$ such that $f(z_n)=z_n$ for all $n \in \mathbb{N}$ and $f(5)=0$. (T/F)
My work: Consider $g(z)=f(z)-z$, clearly $g(z_n)=0$. Thus by identity and uniqueness theorem $g(z)=0 $ for all $z$ in the unit circle, i.e $f(z)=z$, so $f(5)=5$, and the above statement is false.
CHECK the logic.