LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-24

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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.

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Null sequences form an ideal

Statement

The set $N$ of null sequences is an ideal of the ring $C$ of Cauchy sequences (Cauchy sequences form a commutative ring): it is a subgroup under addition, and $c \cdot z \in N$ whenever $c \in C$ and $z \in N$ .

Facts & Assumptions

Given: Null sequences $(z_{n}), (w_{n})$ , a Cauchy sequence $(c_{n})$ , and a rational $ε > 0$ .

[A1]

Null: beyond some index, $∣ z_{n} ∣$ is smaller than any prescribed positive rational.

[L1]

Ordered-field arithmetic in $Q$ (The rationals form a totally ordered field).

[L2]

Triangle inequality and multiplicativity of $∣ \cdot ∣$ (Absolute value and the triangle inequality).

[L3]

Cauchy sequences are bounded (Every Cauchy sequence of rationals is bounded).

[L4]

Null sequences are Cauchy, so $N \subseteq C$ (Null sequences are Cauchy).

Proof

technique · direct

1.1

Fix $N_{z}, N_{w}$ with $∣ z_{n} ∣ < ε / 2$ for $n \geq N_{z}$ and $∣ w_{n} ∣ < ε / 2$ for $n \geq N_{w}$ .

A1L1

1.2

Fix $M \geq 0$ with $∣ c_{n} ∣ \leq M$ ([L3]) and set $B = \max (M, 1) \geq 1$ , so $∣ c_{n} ∣ \leq B$ for all $n$ .

2.1

Sum: for $n \geq \max (N_{z}, N_{w})$ , $∣ z_{n} + w_{n} ∣ \leq ∣ z_{n} ∣ + ∣ w_{n} ∣ < ε$ ; negation: $∣ - z_{n} ∣ = ∣ z_{n} ∣$ . So $N$ is a subgroup under addition.

step 1.1L2

2.2

Fix $N$ with $∣ z_{n} ∣ < ε / B$ for $n \geq N$ .

step 1.2A1L1

3.1

Product: for $n \geq N$ , $∣ c_{n} z_{n} ∣ = ∣ c_{n} ∣ ∣ z_{n} ∣ \leq B ∣ z_{n} ∣ < ε$ ; so $(c_{n} z_{n})$ is null.

step 1.2step 2.2L2

4.1

$N$ is a nonempty additive subgroup of $C$ absorbing multiplication by $C$ : an ideal.

step 2.1step 3.1L4∎

Depends on

Used by

The real numbers Definition
Null rational sequences form a maximal ideal in the ring of rational Cauchy sequences Example
The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences Example
FALSE: the rationals are complete False statement
Free tail ultrafilters and bounded real ultralimit calculus Lemma
The null ideal is maximal Lemma
The reals form a field Theorem

Dependency tree · two levels

12 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., §5.3 (standard reference, not scraped)
L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)