please state the theorem about identity theorem, detailly In today's class, I learned the following theorem:
Let $f$ be analytic on domain $\mathbb{D}$, and {$z_n$} be a sequence of distinct numbers which approaches $z_0$ as $n \to \infty$. Suppose that $f(z_n)=0$ for each $n \geq 1$. Then the function $f$ is identically equal to $0$.
My question is about the mentioned sequence {$z_n$}. Do the conditions of the theorem have to hold for all sequences $z_n \to z_0$, or does there only have to exists some sequence $z_n \to z_0$. I predict that "all sequences" is probably correct.
If you know that the precise meaning of the theorem, please state it in detail.

Let $f$ be analytic function on domain D.

Identitytheorem
  Suppose "there exist point $z_0$ in D and sequence $z_n$ in D" such that $z_n$ converges to $z_0$ satisfying $f(z_n)=0$ for all $n$.
  If  "for infintely many $n$" $z_n$ is not $z_0$ , then $f(z)=0$ in D.

For all sequence $z_n$ converging $z_0$ Is very strong condition.
