Cauchy Sequence Is Eventually Bounded Away From Non-Limit
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Theorem
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Suppose $\sequence {x_n}$ does not converge to $l \in R$.
Then:
- $\exists K \in \N$ and $C \in \R_{>0}: \forall n > K: C < \norm {x_n - l}$
Proof
Because $\sequence {x_n}$ does not converge to $l$:
- $\exists \epsilon \in \R_{>0}: \forall n \in \N: \exists m \ge n: \norm {x_m - l} \ge \epsilon$
Because $\sequence {x_n}$ is a Cauchy sequence:
- $\exists K \in \N: \forall n, m \ge K: \norm {x_n - x_m} < \dfrac \epsilon 2$
Let $M \ge K: \norm {x_M - l} \ge \epsilon$.
Then $\forall n > K$:
\(\ds \epsilon\) | \(\le\) | \(\ds \norm {x_M - l}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \norm {x_M - x_n + x_n - l}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \norm {x_M - x_n} + \norm {x_n - l}\) | Norm Axiom $\text N 3$: Triangle Inequality | |||||||||||
\(\ds \) | \(<\) | \(\ds \dfrac \epsilon 2 + \norm {x_n - l}\) | because $n, M \ge K$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac \epsilon 2\) | \(<\) | \(\ds \norm {x_n - l}\) | subtracting $\dfrac \epsilon 2$ from both sides of the equation |
Let $C = \dfrac \epsilon 2$ and the result follows.
$\blacksquare$