\begin{definition}[Definition:Convergent Sequence/Complex Numbers/Definition 2]
Let $\sequence {z_k} = \sequence {x_k + i y_k}$ be a sequence in $\C$.
$\sequence {z_k}$ '''converges to the limit $c = a + i b$''' {{iff}} 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 $x_n - a$.
\end{definition}