# Convergence of Modulus of Convergent Complex Sequence

## Theorem

Let $\sequence {z_n}$ be a sequence in $\C$.

Let $\sequence {z_n}$ converge to a value $c \in \C$.

Let $\cmod z$ denote the modulus of a complex number $z$.

Then:

$\sequence {\cmod {z_n} }$ converges to a value $\cmod c$.

## Proof

By definition of convergent complex sequence:

$\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$

From the Reverse Triangle Inequality:

$\size {\cmod x - \cmod y} \le \cmod {x - y}$

and the result follows.

$\blacksquare$