lean_workbook_plus_40094

⚠️ Retired — specification defect

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

Let $a=\sqrt{2}$ \n$x_{n+1}-\sqrt{2}=\frac{x_n+2}{x_n+1}-\sqrt{2}= (\sqrt{2}-x_n) \frac{\sqrt{2}-1}{x_n+1} \to |x_{n+1}-\sqrt{2}|< |x_n-\sqrt{2}| \frac{\sqrt{2}-1}{2}$ \nSo $|x_{n+1}-\sqrt{2}|<|x_1-\sqrt{2}|(\frac{\sqrt{2}-1}{2})^n< (\sqrt{2}-1)^n$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_40094  (x : ℕ → ℝ)
  (h₀ : ∀ n, x (n + 1) = (x n + 2) / (x n + 1))
  (h₁ : x 1 = 1) :
  ∀ n, abs (x n - Real.sqrt 2) < (Real.sqrt 2 - 1)^n   :=  by sorry

The recurrence at n=0 requires1=(x₀+2)/(x₀+1), which has no solution. The claimed strict bound also fails at the genuine initial index n=1 by equality.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Set x₁=1 and impose x_(n+1)=(x_n+2)/(x_n+1) only for n≥1. Prove |x_n−√2|≤(√2−1)^n for n≥1, with strict inequality for n≥2, and conclude convergence to√2.

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.