Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition
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Theorem
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Then:
- $\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$
Proof
Let $\epsilon > 0$ be given.
By the definition of a Cauchy sequence:
- $\exists N: \forall n, m > N: \norm {x_n - x_m} < \epsilon$
So
- $\exists N: \forall n > N: \norm {x_{n + 1} - x_n} < \epsilon$
Hence the result follows:
- $\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions, Lemma $3.2.2$