Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-07-24
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Cauchy sequence of rationals
Definition
A sequence of rational numbers is a Cauchy sequence if for every rational there exists such that
Remarks
- The quantifier ranges over rational only. This is deliberate: the definition is stated before the real numbers exist, which is exactly what allows the reals to be constructed from it.
- Informally: the terms of the sequence eventually cluster arbitrarily tightly, without any reference to a limit value.
Depends on
Used by
- The Cauchy sequences of rationals form a commutative ring that is not an integral domain: two eventually-constant sequences with disjoint supports multiply to zero Example
- FALSE: the rationals are complete False statement
- A non-null Cauchy sequence is eventually bounded away from zero, with constant sign Lemma
- Every Cauchy sequence of rationals is bounded Lemma
- Null sequences are Cauchy Lemma
- The rationals embed densely in the reals Lemma
- Cauchy sequences form a commutative ring Theorem
- The reals are complete Theorem
Dependency tree · two levels
5 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
- T. Tao, Analysis I, 3rd ed., §5.1 (standard reference, not scraped)
- W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (standard reference, not scraped)
- Real number (Encyclopedia of Mathematics) (standard reference, not scraped)