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.

Suppose $\lim_{n \to \infty} \norm {x_{n+1} - x_n} = 0$

then:


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

Proof
Let $\epsilon \gt 0$ be given.

By assumption $\exists N \in \N$:
 * (1) $\forall n \gt N: \norm {x_{n+1} - x_n} \lt 0$

Suppose $n, m \gt N$, and $m = n + r > n$, then:

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