False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-31

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.

FALSE: representatives of every $L^{p}$ -Cauchy sequence converge pointwise almost everywhere

Statement

For every Cauchy sequence $(u_{n})$ in $L^{p} (μ)$ and every choice of measurable representatives $f_{n} \in u_{n}$ , the sequence $(f_{n})$ converges pointwise almost everywhere.

Facts & Assumptions

Given: The published typewriter-sequence false statement.

[L1]

The published false statement FALSE: convergence in L^1(mu) forces almost-everywhere convergence supplies a sequence converging in $L^{1}$ but not almost everywhere.

Refutation

Proof technique: Refute in $L^{1}$ by the published typewriter witness: the sequence converges in norm and therefore is Cauchy, but it has no pointwise limit anywhere on $[0, 1]$ .

1.1

The sequence from [L1] converges in $L^{1}$ , hence is Cauchy in $L^{1}$ .

L1given

2.1

The same source records representatives that fail to converge almost [L1, step 1.1] everywhere. So representatives of a Cauchy sequence in $L^{p}$ need not converge pointwise almost everywhere.

3.1

Therefore the universal claim is false.

step 2.1∎

Depends on

FALSE: convergence in L^1(mu) forces almost-everywhere convergence

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Gerald B. Folland, Real Analysis, 2nd ed., Section 2.4, Example (iv) (standard reference, not scraped)
Terence Tao, 245A Notes 4: Modes of convergence, Example 7 (standard reference, not scraped)

FALSE: representatives of every LpLp-Cauchy sequence converge pointwise almost everywhere