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

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

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

Then:


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

Proof
Let $\epsilon > 0$ be given.

By the definition of a Cauchy sequence:


 * $\exists N: \forall n, m > N: \norm {x_n - x_m} < \epsilon$

So


 * $\exists N: \forall n > N: \norm {x_{n + 1} - x_n} < \epsilon$

Hence the result follows:


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