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 Numbers form Non-Archimedean Valued Field

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

it follows that:

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

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$