Alphabeta Math

Equivalent Forms of Completeness

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✓ 17 results · all verified · 15 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 2 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

The five completeness properties of an ordered field: least upper bound, monotone convergence, nested intervals, Bolzano-Weierstrass, and Cauchy completeness

Lemma

Sequence basics in an arbitrary ordered field: limits are unique, limits preserve non-strict inequalities, convergent sequences are Cauchy, Cauchy sequences are bounded, and a Cauchy sequence with a convergent subsequence converges

Lemma

An ordered field with the least-upper-bound property has the nested interval property and is Archimedean

Lemma

Nested intervals plus the Archimedean property imply Bolzano-Weierstrass, by repeated bisection

Lemma

Bolzano-Weierstrass alone forces the Archimedean property, so it needs no separate Archimedean hypothesis

Lemma

Bolzano-Weierstrass implies Cauchy completeness in any ordered field

Lemma

Cauchy completeness plus the Archimedean property imply the monotone convergence property

Lemma

The monotone convergence property alone forces the Archimedean property, so it carries no separate Archimedean hypothesis

Lemma

The monotone convergence property plus the Archimedean property imply the least-upper-bound property

Theorem

For an ordered field the five completeness properties are equivalent, provided the Archimedean property is assumed alongside nested intervals and Cauchy completeness

Definition

The Cesaro means σₙ = (x₀ + … + xₙ)/(n+1) and (C,1)-summability

Theorem

If xₖ → L then σₙ → L: convergence implies (C,1)-summability to the same value

Theorem

Stolz-Cesaro, ∞/∞ form: if bₖ is strictly increasing and unbounded and (aₖ₊₁-aₖ)/(bₖ₊₁-bₖ) → L then aₖ/bₖ → L

Corollary

Stolz-Cesaro, 0/0 form: if bₖ is strictly decreasing to 0, aₖ → 0, and the difference quotient converges, then aₖ/bₖ converges to the same value

Definition

A summability (Toeplitz) matrix, the transformed sequence yₙ = ∑ₖ c_n,k xₖ, and regularity

Theorem

A summability matrix with only finitely many nonzero entries per row is regular iff each column tends to 0, the row sums tend to 1, and the row absolute sums are uniformly bounded

Corollary

The Cesaro matrix satisfies the Silverman-Toeplitz conditions, giving a second proof of the Cesaro mean theorem

Remark

Which of the five completeness properties carry the Archimedean property on their own, and which must be handed it

False statement

FALSE: the nested interval property alone implies the least-upper-bound property

False statement

FALSE: an ordered field in which every Cauchy sequence converges has the least-upper-bound property

False statement

FALSE: if the Cesaro means of a sequence converge then the sequence converges

Sources