Characterisation of Cauchy Sequence in Non-Archimedean Norm

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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 sequence in $R$.

Then:

$\sequence {x_n}$ is a Cauchy sequence if and only if $\displaystyle \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.

Corollary

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Let $\sequence {x_n}$ be a sequence of integers such that:

$\forall n: x_{n + 1} \equiv x_n \pmod {p^n}$


Then:

$\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Proof

Necessary Condition

Let $\epsilon \gt 0$ be given.

By the definition of a Cauchy sequence:

$\exists N: \forall n, m \gt N: \norm {x_n - x_m} \lt \epsilon$

So

$\exists N: \forall n \gt N: \norm {x_{n+1} - x_n} \lt \epsilon$

Hence the result follows:

$\lim_{n \to \infty} \norm {x_{n+1} - x_n} = 0$.

$\Box$

Sufficient Condition

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:

\(\displaystyle \norm {x_m - x_n}\) \(=\) \(\displaystyle \norm {x_{n + r} - x_{n + r - 1} + x_{n + r - 1} - x_{n + r - 2} + \dotsb + x_{n + 1} - x_n}\)
\(\displaystyle \) \(=\) \(\displaystyle \max \set {\norm {x_{n + r} - x_{n + r - 1} }, \norm {x_{n + r - 1} - x_{n + r - 2} }, \dotsc, \norm {x_{n + 1} - x_n} }\) as $\norm {\,\cdot\,}$ is non-Archimedean
\(\displaystyle \) \(=\) \(\displaystyle \norm {x_{n + s} - x_{n + s - 1} }\) for some $s$: $0 < s \le r$
\(\displaystyle \) \(<\) \(\displaystyle \epsilon\) by $(1)$

It follows that:

$\sequence {x_n}$ is a Cauchy sequence.

$\blacksquare$

Sources