Sequence Converges to Within Half Limit/Complex Numbers

Theorem

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

Let $\sequence {z_n}$ be convergent to the limit $l$.

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

Then:

$\exists N: \forall n > N: \cmod {z_n} > \dfrac {\cmod l} 2$

Proof

Suppose $l > 0$.

Let us choose $N$ such that:

$\forall n > N: \cmod {z_n - l} < \dfrac {\cmod l} 2$

Then:

 $\displaystyle \cmod {z_n - l}$ $<$ $\displaystyle \frac {\cmod l} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod l - \cmod {z_n}$ $\le$ $\displaystyle \cmod {z_n - l}$ Reverse Triangle Inequality $\displaystyle$ $<$ $\displaystyle \frac {\cmod l} 2$ $\displaystyle \leadsto \ \$ $\displaystyle \cmod {z_n}$ $>$ $\displaystyle \cmod l - \frac {\cmod l} 2$ $\displaystyle$ $=$ $\displaystyle \frac {\cmod l} 2$

$\blacksquare$