Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

Tails of a convergent improper integral tend to zero

Statement

If ∫a∞f\int_a^\infty f converges, then lim⁡R→∞∫R∞f=0.\lim_{R\to\infty}\int_R^\infty f=0. Equivalently, for every ε>0\varepsilon>0 all sufficiently remote proper tails ∫uvf\int_u^v f have absolute value below ε\varepsilon. The corresponding tails tend to zero at −∞-\infty and at either finite singular endpoint.

Facts & Assumptions

Given: A convergent one-ended improper integral of ff.

[L1]

A convergent integral may be split at every finite truncation (Improper convergence is independent of finite truncations and split points).

[L2]

Its proper remote tails satisfy the Cauchy criterion (Cauchy criterion for improper integrals).

Proof

technique · direct
1.1

Write I=∫a∞fI=\int_a^\infty f. By [L1], ∫R∞f=I−∫aRf\int_R^\infty f=I-\int_a^R f, which tends to zero by the definition of II.

L1
2.1

The epsilon formulation is exactly [L2]. Reversing orientation or replacing infinite truncations by one-sided finite truncations proves all other stated forms.

L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 57 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources