Sequence Converges to Within Half Limit/Normed Division Ring

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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
\(\displaystyle \) \(<\) \(\displaystyle \frac {\norm l} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \norm {x_n}\) \(>\) \(\displaystyle \norm l - \frac {\norm l} 2\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\norm l} 2\)

$\blacksquare$