Definition:Convergent Sequence/Complex Numbers/Definition 2

Definition

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 $x_n - a$.

Note on Domain of $N$

Some sources insist that $N \in \N$ but this is not strictly necessary and can make proofs more cumbersome.

Also see

• Equivalence of Definitions of Convergent Complex Sequence

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.2$. Sequences: Theorem.