Alphabeta Math

Completeness, Completion, and Uniform Continuity

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✓ 15 results · all verified · 11 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

Cauchy sequence in a metric space

Lemma

Every convergent sequence in a metric space is Cauchy

Lemma

Every Cauchy sequence in a metric space is bounded

Lemma

A Cauchy sequence in a metric space with a convergent subsequence converges to that subsequence’s limit

Definition

Complete metric space: every Cauchy sequence converges in the space

Theorem

Closed subspaces of complete metric spaces are complete; the converse under countable choice

Theorem

ℝ and ℝⁿ for n ≥ 1 with the Euclidean metric are complete, componentwise from the Cauchy criterion in ℝ

Theorem

In a complete metric space nested nonempty closed sets whose diameters tend to 0 meet in exactly one point, and this property characterises completeness

Definition

Uniform continuity of a map of metric spaces: one δ serving every point

Definition

Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction

Theorem

Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent

Theorem

A uniformly continuous map sends Cauchy sequences to Cauchy sequences

Theorem

A uniformly continuous map from a dense subspace into a complete metric space extends uniquely to a uniformly continuous map on the whole space

Theorem

A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point

Corollary

The a priori bound d(x^*, xₙ) ≤ qⁿ d(x₁,x₀)/(1-q) and the a posteriori bound d(x^*, xₙ₊₁) ≤ q d(xₙ₊₁,xₙ)/(1-q)

Definition

A completion of a metric space: a complete metric space together with an isometric embedding onto a dense subspace

Theorem

Every metric space has a completion, constructed as the equivalence classes of its Cauchy sequences

Theorem

A completion is unique up to a unique isometry fixing the original space, and uniformly continuous maps into complete spaces extend through it

Lemma

Complete metrizability: admitting a topologically equivalent complete metric is preserved by homeomorphism and by closed subspaces, and (0,∞) has it without being complete

Remark

Completeness belongs to the metric; complete metrizability is characterised on a later page

False statement

FALSE: completeness of a metric space is determined by its topology

False statement

FALSE: every Cauchy sequence in a metric space converges

False statement

FALSE: d(fx, fy) < d(x,y) for all x ≠ y on a complete metric space forces a fixed point

False statement

FALSE: two metrics inducing the same topology have the same Cauchy sequences

Sources