lean_workbook_plus_30981

⚠️ 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.

Prove that for any positive integer $n$, there exists an integer $m$ such that $0 \le \frac{n}{m} - m \le 2$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_30981 (n : ℤ) (hn : n > 0) : ∃ m : ℤ, 0 ≤ n / m - m ∧ n / m - m ≤ 2   :=  by sorry

The source real quotient is integer division in the target, and m=0 is not excluded. A reported positive integer-division witness fixes only the zero-denominator weakness, not the fractional semantics.

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

For every positive integer n, prove ∃m∈ℤ with m>0 and 0≤(n:ℝ)/m−m≤2. A suitable witness is m=⌊√n⌋, using m²≤n≤m²+2m.

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.