Alphabeta Math

Equivalent Forms of Completeness: Examples and Counterexamples

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✓ 10 results · all verified · 8 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
Abstract Algebra
Combinatorics
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

The rational function field ℝ(t) ordered by the eventual sign is an ordered field, worked out

Example

ℚ(√2) carries exactly two distinct field orders, exchanged by the conjugation √2 ↦ -√2

Example

ℝ((t⁻¹)), the formal Laurent series field, is Cauchy complete, non-Archimedean, and lacks the least-upper-bound property

Example

The Cesaro means of (-1)ᵏ converge to 0 although the sequence diverges

Example

Stolz-Cesaro gives (1 + 2 + … + n)/n² → 1/2 and (1ᵖ + … + nᵖ)/nᵖ⁺¹ → 1/(p+1) for natural p

Counterexample

In ℝ(t) the rationals are not dense: no rational lies strictly between 0 and 1/t

Counterexample

On a closed interval of ℚ there is a continuous unbounded function, a bounded one with no maximum, and one without the intermediate value property

Counterexample

Over ℚ there is a nonconstant differentiable function with identically zero derivative, so Rolle and the mean value theorem both fail

Counterexample

aₖ = (-1)ᵏ, bₖ = k have aₖ/bₖ → 0 while the difference quotient oscillates, so Stolz-Cesaro has no converse

Counterexample

A summability matrix failing exactly one Silverman-Toeplitz condition and transforming a convergent sequence to a divergent one

Sources