Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary

Theorem

Let $\struct {R, \norm{\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$

Let $\sequence {x_n}$ be a Cauchy sequence such that $\sequence {x_n}$ does not converge to $0_R$.

Then:

$\exists N \in \N: \forall n, m \ge N: \norm {x_n} = \norm {x_m}$

Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

Let $\sequence {x_n}$ be a Cauchy sequence such that $\sequence {x_n}$ does not converge to $0$.

Then:

$\exists N \in \N: \forall n, m \ge N: \norm {x_n}_p = \norm {x_m}_p$

$\exists N_1 \in \N$ and $C \in \R_{\gt 0}: \forall n \ge N_1: \norm {x_n} \gt C$

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

$\exists N_2 \in \N: \forall n, m \ge N_2: \norm {x_n - x_m} < C$

Let $N = \max \set {N_1, N_2}$.

Let $n, m \ge N$.

Then:

$\norm {x_n - x_m} < C < \norm {x_n}$

$\norm {x_n} = \norm {x_m}$

The result follows.

$\blacksquare$