Alphabeta Math

Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness: Examples and Counterexamples

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✓ 11 results · all verified · 6 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 5 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

Example

The Babylonian sequence x₁ = 2, xₖ₊₁ = (xₖ + 2/xₖ)/2 decreases to √2

Example

The sequence x₁ = 1, xₖ₊₁ = √2 + xₖ increases to 2

Example

The nested intervals [0, 1/k] intersect in exactly {0}

Example

The sequence (-1)ᵏ(1 + 1/k) is bounded with subsequential limit set exactly {-1, 1}

Example

The sequence xₖ₊₁ = (xₖ + 1)/3 is contractive with c = 1/3 and converges to 1/2

Counterexample

The truncated decimal approximations of √2 form a Cauchy sequence of rationals with no rational limit

Counterexample

The nested open intervals (0, 1/k) have empty intersection

Counterexample

The nested closed unbounded sets [k, ∞) have empty intersection, so boundedness cannot be dropped

Counterexample

xₖ = √k has xₖ₊₁ - xₖ → 0 and is not Cauchy

Counterexample

The sequence 1, 1, 2, 1, 3, 1, 4, … is unbounded and has a convergent subsequence

Counterexample

xₖ₊₁ = xₖ + 1/xₖ from x₁ = 1 has strictly decreasing consecutive gaps and diverges, so no uniform c < 1 exists

Sources