DefinitionDefinition: AI-adaptedProof: 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.

Null sequence

Definition

A sequence $(a_{n})_{n \geq 1}$ of rational numbers is a null sequence if for every rational $ε > 0$ there exists $N \in N$ such that

$∣ a_{n} ∣ < ε for all n \geq N .$

Remarks

Equivalently: $(a_{n})$ converges to $0$ (with rational tolerances).
In the construction of $R$ , two Cauchy sequences will be declared equivalent exactly when their difference is a null sequence; the null sequences are the "zero" of the construction.

Depends on

Used by

The sequence 1/n is null Example
FALSE: the rationals are complete False statement
A non-null Cauchy sequence is eventually bounded away from zero, with constant sign Lemma
Free tail ultrafilters and bounded real ultralimit calculus Lemma
Null sequences are Cauchy Lemma
Null sequences form an ideal Lemma
The null ideal is maximal Lemma
The reals form a field Theorem
The reals form a totally ordered field 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., §5.2 (standard reference, not scraped)
Real number (Encyclopedia of Mathematics) (standard reference, not scraped)