# Sequence Converges to Within Half Limit/Normed Division Ring

## Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$.

Let $\sequence {x_n}$ be a sequence in $R$.

Let $\sequence {x_n}$ be convergent in the norm $\norm {\, \cdot \,}$ to the following limit:

$\displaystyle \lim_{n \mathop \to \infty} x_n = l \ne 0$

Then:

$\exists N: \forall n > N: \norm {x_n} > \dfrac {\norm l} 2$

## Proof

Since $l \ne 0$, by norm axiom (N1): $\norm l > 0$.

Let us choose $N$ such that:

$\forall n > N: \norm {x_n - l} < \dfrac {\norm l} 2$

Then:

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

$\blacksquare$