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 $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$
- Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
- Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric
- Convergence of a sequence in a metric space: $x_k \to x$ iff $d(x_k, x) \to 0$ in $\mathbb{R}$
- Cauchy sequence in a metric space
- Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
Used by
- f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
- Complex series, absolute convergence, complex power series, and radius of convergence Definition
- The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts Theorem
- There is no continuous logarithm on all of ℂ∖{0} Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 124 results over 22 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
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)