lean_workbook_plus_69897

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $\alpha$ be a real number, $1<\alpha<2$ . Show that $\alpha$ can uniquely be represented as the infinite product $\alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots$ with $n_i$ positive integers satisfying $n_i^2\le n_{i+1}$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_69897 (α : ℝ) (hn : 1 < α ∧ α < 2) : ∃! n : ℕ → ℕ, (∀ i : ℕ, 0 < n i ∧ (n i) ^ 2 ≤ n (i + 1)) ∧ α = ∏' i : ℕ, (1 + 1 / n i)   :=  by sorry

Compute each factor in ℝ, e.g.1+1/(n i:ℝ), before taking the infinite product; retain positive integer denominators and the growth condition.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Compute each factor in ℝ, e.g.1+1/(n i:ℝ), before taking the infinite product; retain positive integer denominators and the growth condition.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.