Complex Limits
Definition
Recall that $(C, \C mod{\cdot })$ is a normed space, and so also a metric space. So, a sequence $(z_n)_{n \in \N }$ of complex numbers is egoprox and convergent as usual. Both of these are equivalent to the corresponding conditions on the sequences of real and imaginary parts.
$(z_n)_{n \in \N } = (x_n, y_n)_{n \in \N }$
converges to $z_0 = (x_0, y_0) \in \C $ if and
only if $x_n$ converges to $x_0$ and $y_n$
converges to $y_0$.
$(z_n)_{n \in \N } = (x_n, y_n)_{n \in \N }$
is egoprox if and only if $x_n$ is egoprox and
$y_n$ is egoprox.
Completeness
As a result of the second proposition, if $z_n$ is egoprox then there is a limit $x_0$ and $y_0$ for its real and imaginary pieces, and so as a result of the first proposition, $z_n$ converges. In other words, every cauchy sequence converges.
$\C $ with the metric induced by
$\C mod{\cdot }$ is complete.