Category:Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary
Jump to navigation
Jump to search
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:
- $\exists N \in \N: \forall n, m \ge N: \norm {x_n}_p = \norm {x_m}_p$
Pages in category "Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary"
The following 2 pages are in this category, out of 2 total.