# 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 $\displaystyle \lim_{n \mathop \to \infty} z_n = l$ where $l \ne 0$.

Then:

$\exists N: \forall n > N: \left\vert{z_n}\right\vert > \dfrac {\left\vert{l}\right\vert} 2$

## Proof

Suppose $l > 0$.

Let us choose $N$ such that:

$\forall n > N: \left\vert{z_n - l}\right\vert < \dfrac {\left\vert{l}\right\vert} 2$

Then:

 $\displaystyle \left\vert{z_n - l}\right\vert$ $<$ $\displaystyle \frac {\left\vert{l}\right\vert} 2$ $\displaystyle \implies \ \$ $\displaystyle \left\vert{l}\right\vert - \left\vert{z_n}\right\vert$ $\le$ $\displaystyle \left\vert{z_n - l}\right\vert$ Reverse Triangle Inequality $\displaystyle$ $<$ $\displaystyle \frac {\left\vert{l}\right\vert} 2$ $\displaystyle \implies \ \$ $\displaystyle \left\vert{z_n}\right\vert$ $>$ $\displaystyle \left\vert{l}\right\vert - \frac {\left\vert{l}\right\vert} 2$ $\displaystyle$ $=$ $\displaystyle \frac {\left\vert{l}\right\vert} 2$

$\blacksquare$