1.2 Convergence · Preliminaries to Complex Analysis

[1.2.1]Definition(Convergence of Sequences)#

A sequence $\{z_n\} \subset \mathbb{C}$ converges to $w \in \mathbb{C}$ if:

\lim_{n \to \infty} |z_n - w| = 0

We write $w = \lim_{n \to \infty} z_n$ .

[1.2.2]Proposition(Component-wise Convergence)#

A sequence $\{z_n\} \subset \mathbb{C}$ converges to $w$ if and only if the sequences of real and imaginary parts of $z_n$ converge to the real and imaginary parts of $w$ , respectively.

[1.2.3]Definition(Cauchy Sequence)#

A sequence $\{z_n\} \subset \mathbb{C}$ is a Cauchy sequence if for every $\varepsilon > 0$ , there exists an integer $N > 0$ such that:

|z_n - z_m| < \varepsilon \quad \text{whenever } n, m > N

[1.2.4]Theorem(Completeness of

\mathbb{C}

)#

The complex numbers $\mathbb{C}$ form a complete metric space. That is, every Cauchy sequence in $\mathbb{C}$ converges to a limit in $\mathbb{C}$ .

Proof.

Since $\{z_n\}$ is Cauchy if and only if the component sequences $\{\text{Re}(z_n)\}$ and $\{\text{Im}(z_n)\}$ are Cauchy in $\mathbb{R}$ , completeness of $\mathbb{C}$ follows directly from the completeness of $\mathbb{R}$ .

1.2Convergence