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

This category contains pages concerning Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary:

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:

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

Corollary

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:

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