Equivalence of Definitions of Convergent Complex Sequence

$(1)$ 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:

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

$(2)$ 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 $\epsilon' \in \R_{>0}$ such that $\epsilon' = \dfrac \epsilon 2$

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

Then:

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