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:
\(\ds \sqrt {\paren {x_n - a}^2 + \paren {y_n - b}^2}\) | \(<\) | \(\ds \epsilon\) | Definition of Complex Modulus | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x_n - a}^2 + \paren {y_n - b}^2\) | \(<\) | \(\ds \epsilon^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x_n - a}^2\) | \(<\) | \(\ds \epsilon^2\) | |||||||||||
\(\ds \paren {y_n - b}^2\) | \(<\) | \(\ds \epsilon^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {x_n - a}\) | \(<\) | \(\ds \epsilon\) | |||||||||||
\(\ds \size {y_n - b}\) | \(<\) | \(\ds \epsilon\) |
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:
\(\ds \cmod {z_n − c}\) | \(=\) | \(\ds \cmod {x_n + i y_n − \paren {a + i b} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {x_n − a + i \paren {y_n − b} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \cmod {x_n − a} + \cmod {i \paren {y_n − b} }\) | Triangle Inequality for Complex Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {x_n − a} + \cmod {y_n − b}\) | Definition of Complex Modulus | |||||||||||
\(\ds \) | \(<\) | \(\ds \frac \epsilon 2 + \cmod {y_n − b}\) | as $\cmod {x_n − a} < \dfrac \epsilon 2$ | |||||||||||
\(\ds \) | \(<\) | \(\ds \frac \epsilon 2 + \frac \epsilon 2\) | as $\cmod {y_n − b} < \dfrac \epsilon 2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \epsilon\) |
Thus $\sequence {z_n}$ is a convergent complex sequence by definition 1.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Theorem.