Equivalence of Definitions of Convergent Complex Sequence

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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.

$\blacksquare$


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