# Uniqueness theorem for functions with poles?

I am trying to solve the following question:

Prove the uniqueness theorem for functions with poles: If $f$ is holomorphic on $D$ apart from some poles, and there exists $z_n \to z$ in $D$ such that $f(z_n) = 0$ then $f$ is constant.

There are two things I don't understand:

1. If the function is constant, how can it have poles? By definition, poles are points where the function approaches infinity.
2. Assuming it is possible, can I prove it by removing small circle around each pole such that $z$ is still in the remaining area and using the regular uniqueness theorem?

Thanks!

Regarding the second point, you need to exclude the possibility that $z$ is a pole, then you can use the ordinary identity theorem to conclude that $f$ is constant on the connected component of $D\setminus P$ containing $z$, where $P$ is the (possibly empty) set of poles of $f$. Then - assuming $D$ is connected, otherwise $f$ need not be constant - find that $D\setminus P$ is still connected, hence $f$ is constant on $D\setminus P$, whence $P = \varnothing$.