lean_workbook_plus_37756

⚠️ 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.

Given a sequence of integers $(x_n)_{n=1}^{\infty}$ satisfying $x_1 = 1$ and $x_n < x_{n+1} \le 2n$ for all $n$, prove that for every positive integer $k$ there exist indices $r, s$ such that $x_r-x_s = k$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_37756 (x : ℕ → ℤ) (hx: x 1 = 1 ∧ ∀ n, x n < x (n+1) ∧ x (n+1) ≤ 2*n) : ∀ k, ∃ r s, x r - x s = k   :=  by sorry

The zero instance of the formal upper bound contradicts x₁=1.

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

Restrict the sequence inequalities to n≥1 and conclude that for every positive k there are positive indices r,s with x_r−x_s=k.

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.