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

Proof
By Cauchy Sequence Is Eventually Bounded Away From Non-Limit then:
 * $\exists N_1 \in \N$ 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:
 * $\exists N_2 \in \N: \forall n, m \ge N_2: \norm{x_n - x_m} \lt C$

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

Let $n, m \ge N$.

Then:
 * $\norm{x_n - x_m} \lt C \lt \norm {x_n}$

By Corollary to Three Points in Ultrametric Space have Two Equal Distances then:
 * $\norm {x_n} = \norm {x_m}$

The result follows.