An invariant interval for a quartic rational recurrence

Proved

WorkbookCorrected.plus_16969

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

corrected-formalizationlean-workbooksource-checked

Define a sequence $\{x_n\}$ by

x_1=2,x_{n+1}=\frac{x_n^4+9}{10x_n}.

Prove that $\frac45<x_n\leq\frac54$ for all $n>1$

Formalization Note: The recurrence is restricted to n≥1, matching the source initial index. The full interval bound is proved for every n≥2.

Source: InternLM Lean-Workbook, record lean_workbook_plus_16969 (Apache-2.0).

Preamble

import Mathlib

Formal statement

theorem WorkbookCorrected.plus_16969 (x : ℕ → ℝ) (hx : x 1=2)
    (h : ∀ n : ℕ, 1≤n → x (n+1)=(x n^4+9)/(10*x n)) :
    ∀ n : ℕ, 2≤n → 4/5<x n ∧ x n≤5/4 := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_16969; Apache-2.0

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