TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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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.

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The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts

Statement

The metric space $\mathbb C$ is complete. A sequence $z_n=x_n+iy_n$ converges to $x+iy$ exactly when $x_n\to x$ and $y_n\to y$ . The conventions and prerequisite facts used below are recorded in The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane, For $n \ge 1$ a sequence in $\mathbb{R}^n$ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and $\mathbb{R}^n$ is complete in every norm.

Facts & Assumptions

Given: A complex sequence $z_n=x_n+iy_n$ .

Proof

technique · direct

1.1

The complex metric is the Euclidean metric on $\mathbb R^2$ .

given

2.1

Apply componentwise convergence and completeness in the published Euclidean-space theorem.

given∎

Depends on

Used by

The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 105 results over 20 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)