if it exist the limit of $f(z_n)$ then there exist the the limit of $z_n$ Let $D \subset \mathbb{C}$ a closed compact subset of the complex plane.
Let $f:D \to \mathbb{C}$ a continuous function.
Let $\{z_n\}_{n \in \mathbb{N}} \in D$ a sequence of $D$.
Is it true that
$$
\lim_{n \to \infty} f(z_n)=L \Longrightarrow \lim_{n \to \infty} z_n = z_0 \in D
$$
Thanks

No.  Take $f$ a constant function and $\{z_n\}$ any divergent sequence.
