Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition
Jump to navigation
Jump to search
Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $\ds \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.
Then:
- $\sequence {x_n}$ is a Cauchy sequence.
Proof
Let $\epsilon > 0$ be given.
By assumption $\exists N \in \N$ such that:
- $(1) \quad \forall n > N: \norm {x_{n + 1} - x_n} < 0$
Suppose $n, m > N$, and $m = n + r > n$.
Then:
\(\ds \norm {x_m - x_n}\) | \(=\) | \(\ds \norm {x_{n + r} - x_{n + r - 1} + x_{n + r - 1} - x_{n + r - 2} + \dotsb + x_{n + 1} - x_n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\norm {x_{n + r} - x_{n + r - 1} }, \norm {x_{n + r - 1} - x_{n + r - 2} }, \dotsc, \norm {x_{n + 1} - x_n} }\) | as $\norm {\,\cdot\,}$ is non-Archimedean | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {x_{n + s} - x_{n + s - 1} }\) | for some $s$: $0 < s \le r$ | |||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) | by $(1)$ |
It follows that:
- $\sequence {x_n}$ is a Cauchy sequence.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions, Lemma $3.2.2$