ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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A standard p-adic digit sequence is Cauchy and converges in Zp

Example

The sequence

$t_{n} : = 1 + p + \dots + p^{n - 1} \in Z \subseteq Z_{p}$

is Cauchy for the additive $p$ -adic metric and converges in $Z_{p}$ .

Facts & Assumptions

Given: The sequence $t_{n} = 1 + p + \dots + p^{n - 1}$ in $Z_{p}$ .

[F1]

The $p$ -adic metric measures how many initial residue coordinates agree (The p-adic metric on Zp is determined by the first coordinate at which two compatible residue systems differ).

[L1]

$Z_{p}$ is complete for that metric (Zp is Hausdorff, totally disconnected, and complete, and compact assuming Choice).

Verification

technique · direct

1.1

If $m > n$ , then $t_{m} - t_{n} = p^{n} (1 + p + \dots + p^{m - n - 1})$ , so the first $n$ residue coordinates of $t_{m}$ and $t_{n}$ agree. By [F1], this gives $d_{p} (t_{m}, t_{n}) \leq p^{- n}$ . Hence $(t_{n})$ is Cauchy.

F1givenalgebra

2.1

Since $Z_{p}$ is complete by [L1], the Cauchy sequence $(t_{n})$ converges to some element of $Z_{p}$ .

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jordan Bell, Explicit construction of the p-adic numbers (standard reference, not scraped)