Can an holomorphic function have a convergent sequence of points in its domain with only real values?

Question

Let $D \subset \mathbb{C}$ be a open subset of the complex plane

Let $f: D \to \mathbb{C}$ be a holomorphic function

Let $\{z_n\}_{n \in \mathbb{N}} \subset D$ be a sequence in $D$ such that $z_n \to s \in D$ and we have $f(z_n) \to f(s)=0$

My question is: Is there a subsequence $\{z_{n_k}\}_{k \in \mathbb{N}}$ such that $\forall k \in \mathbb{N}: \Im(f(z_{n_k})) \neq 0$

Thanks.

Answer 1

Well... Not necessarily. Let $f(z)=z$ and $z_n=1/n$ for example.