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:

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

Then:

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

Proof

Since $l \ne 0$, by Norm Axiom $\text N 1$: Positive Definiteness:

$\norm l > 0$

Let us choose $N$ such that:

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

Then:

\(\ds \norm {x_n - l}\)

\(<\)

\(\ds \frac {\norm l} 2\)

\(\ds \leadsto \ \ \)

\(\ds \norm l - \norm {x_n}\)

\(\le\)

\(\ds \norm {x_n - l}\)

Reverse Triangle Inequality

\(\ds \)

\(<\)

\(\ds \frac {\norm l} 2\)

\(\ds \leadsto \ \ \)

\(\ds \norm {x_n}\)

\(>\)

\(\ds \norm l - \frac {\norm l} 2\)

\(\ds \)

\(=\)

\(\ds \frac {\norm l} 2\)

$\blacksquare$