# Equivalence of Definitions of Convergent Complex Sequence

## Theorem

The following definitions of the concept of Convergent Complex Sequence are equivalent:

### Definition 1

Let $\sequence {z_k}$ be a sequence in $\C$.

$\sequence {z_k}$ converges to the limit $c \in \C$ if and only if:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$

where $\cmod z$ denotes the modulus of $z$.

### Definition 2

Let $\sequence {z_k} = \sequence {x_k + i y_k}$ be a sequence in $\C$.

$\sequence {z_k}$ converges to the limit $c = a + i b$ if and only if both:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$

where $\size {x_n - a}$ denotes the absolute value of $x_n - a$.

## Proof

### $(1)$ implies $(2)$

Let $\sequence {z_n}$ be a convergent complex sequence by definition 1.

Then by definition:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$

Let $z_n = x_n + i y_n$.

Let $c = a + i b$.

Let $\epsilon \in \R_{>0}$.

Let $N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$.

Then:

 $\displaystyle \sqrt {\paren {x_n - a}^2 + \paren {y_n - b}^2}$ $<$ $\displaystyle \epsilon$ Definition of Complex Modulus $\displaystyle \leadsto \ \$ $\displaystyle \paren {x_n - a}^2 + \paren {y_n - b}^2$ $<$ $\displaystyle \epsilon^2$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {x_n - a}^2$ $<$ $\displaystyle \epsilon^2$ $\displaystyle \paren {y_n - b}^2$ $<$ $\displaystyle \epsilon^2$ $\displaystyle \leadsto \ \$ $\displaystyle \size {x_n - a}$ $<$ $\displaystyle \epsilon$ $\displaystyle \size {y_n - b}$ $<$ $\displaystyle \epsilon$

Thus $\sequence {z_n}$ is a convergent complex sequence by definition 2.

$\Box$

### $(2)$ implies $(1)$

Let $\sequence {z_n} = \sequence {x_n + y_n}$ be a convergent complex sequence by definition 2.

Then by definition:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \size {x_n - a} < \epsilon \text { and } \size {y_n - b} < \epsilon$

where $a + i b = c$.

Let $\epsilon \in \R_{>0}$.

Let $N \in \R: n > N \implies \size {x_n - a} < \dfrac \epsilon 2 \text { and } \size {y_n - b} < \dfrac \epsilon 2$.

Then:

 $\displaystyle \cmod {z_n − c}$ $=$ $\displaystyle \cmod {x_n + i y_n − \paren {a + i b} }$ $\displaystyle$ $=$ $\displaystyle \cmod {x_n − a + i \paren {y_n − b} }$ $\displaystyle$ $\le$ $\displaystyle \cmod {x_n − a} + \cmod {i \paren {y_n − b} }$ Triangle Inequality for Complex Numbers $\displaystyle$ $=$ $\displaystyle \cmod {x_n − a} + \cmod {y_n − b}$ Definition of Complex Modulus $\displaystyle$ $<$ $\displaystyle \frac \epsilon 2 + \cmod {y_n − b}$ as $\cmod {x_n − a} < \dfrac \epsilon 2$ $\displaystyle$ $<$ $\displaystyle \frac \epsilon 2 + \frac \epsilon 2$ as $\cmod {y_n − b} < \dfrac \epsilon 2$ $\displaystyle$ $=$ $\displaystyle \epsilon$

Thus $\sequence {z_n}$ is a convergent complex sequence by definition 1.

$\blacksquare$