Sequence Converges to Within Half Limit/Complex Numbers

Theorem
Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.

Let $\left \langle {z_n} \right \rangle$ be convergent to the limit $l$.

That is, let $\displaystyle\lim_{n \to \infty} z_n = l$ where $l \ne 0$.

Then:
 * $\displaystyle\exists N: \forall n > N: \left\vert{z_n}\right\vert > \frac {\left\vert{l}\right\vert} 2$

Proof
Suppose $l > 0$.

Let us choose $N$ such that $\displaystyle \forall n > N: \left\vert{z_n - l}\right\vert < \frac {\left\vert{l}\right\vert} 2$.

Then: