Characterisation of Cauchy Sequence in Non-Archimedean Norm

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 $R$.

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


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