Convergent Sequences of Complex Numbers

In the setting of Complex Analysis: Standing Notation, let $(z_{n})$ be a sequence in $\mathbb{C}$, and let $w\in\mathbb{C}$.

(i) Convergence to a point. $(z_{n})$ converges to $w$, written $z_{n}\to w$, if for every real $\varepsilon>0$ there is a natural number $N$ with

$$|z_{n}-w|<\varepsilon\qquad\text{for every }n\in\mathbb{N}\text{ with }n\ge N.$$

(ii) Convergent sequences. $(z_{n})$ is convergent if it converges to some complex number.