Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition

Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$.

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

Let $\displaystyle \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.

Then:


 * $\sequence {x_n}$ is a Cauchy sequence.

Proof
Let $\epsilon > 0$ be given.

By assumption $\exists N \in \N$ such that:
 * $(1) \quad \forall n > N: \norm {x_{n + 1} - x_n} < 0$

Suppose $n, m > N$, and $m = n + r > n$.

Then:

It follows that:
 * $\sequence {x_n}$ is a Cauchy sequence.