lean_workbook_plus_55284

⚠️ 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.

We define a new sequence $b_n=1-\frac{1}{4b_{n-1}}$ , where $b_{0}=1.$ Then we show with induction that $\frac12 \leq b_{n} < b_{n-1} \leq 1$ and determine that $lim_{n \to \infty}b_{n}=\frac12.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_55284 (b : ℕ → ℝ) (h₁ : b 0 = 1) (h₂ : ∀ n, b (n + 1) = 1 - 1 / (4 * b n)) : ∀ n, 1 / 2 ≤ b n ∧ b n < b (n - 1) ∧ b (n - 1) ≤ 1   :=  by sorry

Strict decrease includes n=0 and demands b₀<b₀ after truncated subtraction.

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

Proposed corrected statement

State the bound1/2≤b_n<b_(n−1)≤1 only for n≥1, and separately prove b_n→1/2 for the stated recurrence.

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.