lean_workbook_plus_32532

⚠️ 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.

Prove $(1+\dfrac{1}{n})^k<1+\dfrac{k}{n}+\dfrac{k^2}{n^2}$ ,for all $k \le n,n \in N$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_32532 : ∀ k n : ℕ, k ≤ n → (1 + 1 / n)^k < 1 + k / n + k^2 / n^2   :=  by sorry

The binomial comparison uses natural division and includes k=0, where the intended strict inequality is equality.

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

Proposed corrected statement

Use real ratios and require 1≤k≤n; alternatively state the non-strict inequality for 0≤k≤n with n≥1.

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.