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

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## 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 $\Q_p$