lean_workbook_plus_82553

⚠️ 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.

Calculate $\mathop {\lim }\limits_{n \to \infty } \frac{{\sqrt {n^3 - 2n^2 + 1} + \sqrt[3]{{n^4 + 1}}}}{{\sqrt[4]{{n^6 + 6n^5 + 2}} - \sqrt[5]{{n^7 + 3n^3 + 1}}}}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_82553 : ∀ ε : ℝ, ε > 0 → ∃ N : ℕ, ∀ n : ℕ, n ≥ N → |(Real.sqrt (n ^ 3 - 2 * n ^ 2 + 1) + (n ^ 4 + 1) ^ (1 / 3)) / (( n ^ 6 + 6 * n ^ 5 + 2) ^ (1 / 4) - (n ^ 7 + 3 * n ^ 3 + 1) ^ (1 / 5)) - 1| < ε   :=  by sorry

Use real nth-root exponents1/3,1/4,1/5 and cast the bases to ℝ before subtraction and division in the sequence; retain the intended limit1.

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

Confirmed directly from the Lean text: the numeric-literal exponents carry no type ascription, so they elaborate at type ℕ where division truncates (^ (1 / 3) → ^0; ^ (1 / 4) → ^0; ^ (1 / 5) → ^0).

Proposed corrected statement

Use real nth-root exponents1/3,1/4,1/5 and cast the bases to ℝ before subtraction and division in the sequence; retain the intended limit1.

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.