Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm

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

Proof
From:
 * P-adic Norm is Non-Archimedean Norm
 * Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary

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