Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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.

Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary

Definition

Let X be a set and let fn,f:X→C. The sequence (fn) converges uniformly to f when ∀ε>0 ∃N ∀n≥N ∀x∈X:∣fn(x)−f(x)∣<ε. It is uniformly Cauchy when ∀ε>0 ∃N ∀m,n≥N ∀x∈X:∣fm(x)−fn(x)∣<ε.

Writing fn=un+ivn and f=u+iv, uniform convergence in complex modulus is equivalent to uniform convergence of both real component sequences. This follows from ∣un−u∣,∣vn−v∣≤∣fn−f∣ and ∣fn−f∣≤∣un−u∣+∣vn−v∣. These are the complex analogues of Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions, using the metric of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane and the componentwise convergence clause in The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 67 results over 19 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