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: