Alphabeta MathHow 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.
The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
Definition
For and , put Under the identification , this is exactly the metric induced by the Euclidean norm of The -norms for rational , and . It is a metric by Each is a norm on , and the induced metrics are exactly , and of the published metric-spaces page, so the metric axioms are established rather than assumed.
Convergence in , Cauchy sequences in , and continuity of maps between subsets of mean the notions of Convergence of a sequence in a metric space: iff in , Cauchy sequence in a metric space, and Continuity of a map between metric spaces, at a point and globally, in the - form for (and the restricted metric on a subset). These uses are therefore licensed by the metric-space definition Metric space: iff , symmetry, and the triangle inequality; pseudometric and ultrametric.
Depends on
Real and imaginary parts, complex conjugation, and modulus
The -norms for rational , and
Each is a norm on , and the induced metrics are exactly , and of the published metric-spaces page
Metric space: iff , symmetry, and the triangle inequality; pseudometric and ultrametric
Convergence of a sequence in a metric space: iff in
Cauchy sequence in a metric space
Continuity of a map between metric spaces, at a point and globally, in the - form