# Cauchy Sequence Is Eventually Bounded Away From Zero

This article has been proposed for deletion. In particular: This has been superseded by Cauchy Sequence Is Eventually Bounded Away From Non-Limit |

## Theorem

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

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

Suppose $\sequence {x_n}$ does not converge to $0$, then:

- $\exists K \in \N$ and $C \in \R_{> 0}: \forall n > K: C < \norm {x_n}$

## Proof

Since $\sequence {x_n}$ does not converge to $0$ then:

- $\exists \epsilon \in \R_{> 0}: \forall n \in \N, \exists m \ge n: \norm {x_m} \ge \epsilon$

Since $\sequence {x_n}$ is a Cauchy sequence then:

- $\exists K \in \N: \forall n, m \ge K: \norm {x_n - x_m} < \dfrac \epsilon 2$

Let $M \ge K: \norm {x_M} \ge \epsilon$

Then $\forall n > K$:

\(\displaystyle \epsilon\) | \(\le\) | \(\displaystyle \norm {x_M}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \norm {x_M - x_n + x_n}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \norm {x_M - x_n } + \norm {x_n }\) | $\quad$ Axiom (N3) of norm (Triangle Inequality) | $\quad$ | |||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle \dfrac \epsilon 2 + \norm {x_n }\) | $\quad$ Since $n, M \ge K$ | $\quad$ | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \dfrac \epsilon 2\) | \(<\) | \(\displaystyle \norm {x_n }\) | $\quad$ Subtracting $\dfrac \epsilon 2$ from both sides of the equation. | $\quad$ |

Let $C = \dfrac \epsilon 2$ and the result follows.

$\blacksquare$