lean_workbook_plus_59824

⚠️ 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 that $x_n<n~\forall~n \in \mathbb{N}$, which is equivalent to $\sqrt{1+nx_{n-1}}<n \Leftrightarrow x_{n-1}< \frac{n^2-1}{n} \Leftrightarrow x_{n}< \frac{n(n+2)}{n+1}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_59824 (x : ℕ → NNReal) (hx: x 0 = 1) (hn: ∀ n, x (n + 1) = Real.sqrt (1 + n * x n)) : ∀ n, x n < n   :=  by sorry

The excerpt leaves the initial value unstated, so the intended source-indexed version needs clarification.

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

Proposed corrected statement

Restore the source recurrence x(n)=sqrt(1+n*x(n−1)) and its actual initial value and starting index before asserting x(n)<n. If keeping the posted recurrence, restrict the bound to n≥2.

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.