sequence of complex numbers

Definition

If there no extra statement, we assume that $a_{n} \in C$

A sequence of complex numbers is a function from $Z_{\geq m}$ to $C$ , denoted as $(a_{n})_{n \geq m}$
$(a_{n})_{n \geq m}$ converges to $L \in C$ , we write $a_{n} \to L$ iff $\forall ϵ > 0, \exists N \in Z_{\geq m}, \forall n \geq N, d (a_{n} - L) < ϵ$ , L is called the limit of the sequence
$(a_{n})_{n \geq m}$ is a Cauchy sequence iff $\forall ϵ > 0, \exists N \in Z_{\geq m}, \forall n, m \geq N, d (a_{n} - a_{m}) < ϵ$
$L \in \overline{R}$ is a limit point of $(a_{n})_{n \geq m}$ iff $\forall ϵ > 0, \exists N \in Z_{\geq m}, \exists n \geq N, d (a_{n} - L) < ϵ$

The limit of a sequence is unique, because $C$ is a metric space,thus a Hausdorff space, so the limit of a sequence is unique
if $(a_{n})_{n \geq m}$ converges to $x$ $⟺$ $\forall k \in Z_{\geq m}$ , $(a_{n})_{n \geq k}$ converges to $x$
if $a_{n}, b_{n}$ converges to $x, y$ respectively, then the following statements are true
- $a_{n} + b_{n}$ converges to $x + y$
- $a_{n} b_{n}$ converges to $x y$
- if $y \neq = 0 \land b_{n} \neq = 0$ , then $\frac{1}{b _{n}}$ converges to $\frac{1}{y}$
- if $y \neq = 0 \land b_{n} \neq = 0$ , then $\frac{a _{n}}{b _{n}}$ converges to $\frac{x}{y}$
A convergent sequence has only one limit point, which is the limit of the sequence
$a_{n} \to 0$ $⟺$ $∣ a_{n} ∣ \to 0$
$a_{n} + i b_{n} \to x + i y$ $⟺$ $a_{n} \to x \land b_{n} \to y$