lean_workbook_plus_64557

⚠️ 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.

Find the limit of the sequence $x_{n}$ defined as $x_{1}=\dfrac{1}{3} , x_{2n}=\dfrac{1}{3}x_{2n-1} , x_{2n+1}=\dfrac{1}{3}+x_{2n}, n=1,2,\cdots$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_64557 (x : ℕ → ℝ) (x1 : x 0 = 1 / 3) (x2n : ∀ n, x (2 * n) = 1 / 3 * x (2 * n - 1)) (x2np1 : ∀ n, x (2 * n + 1) = 1 / 3 + x (2 * n)) : ∃ l, ∀ ε > 0, ∃ N, ∀ n ≥ N, |x n - l| < ε   :=  by sorry

The initial value was shifted to x₀ while the recurrence is imposed at zero, giving1/3=1/9. The genuine positive-index sequence has two different subsequential limits.

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=1/3 and impose x_(2n)=x_(2n−1)/3, x_(2n+1)=1/3+x_(2n) only for n≥1. Prove the odd subsequence tends to1/2 and the even subsequence to1/6; conclude that the full sequence has no limit.

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.