lean_workbook_plus_39493

⚠️ 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.

Find the limit of the sequence $(a_n)_{n\ge 1}$ defined by $a_{n+1} = \sqrt{a_n + 2}$, where $a_1 > 2$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_39493 (a : ℕ → ℝ) (a1 : a 0 > 2) (a_rec : ∀ n, a (n + 1) = Real.sqrt (a n + 2)) : ∃ n, ∀ ε > 0, |a n - 2| < ε   :=  by sorry

Convergence was replaced by a single index working for every ε, equivalent to reaching2 exactly.

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

Proposed corrected statement

Prove aₙ→2, i.e. ∀ε>0 ∃N ∀n≥N, |aₙ−2|<ε, for a₀>2 and aₙ₊₁=√(aₙ+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.