Lean-Workbook Plus 13953: Logarithmic inequality

Proved

WorkbookRestored.plus_13953

by wamlart · Sep 12, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

lean-workbookrestored-importssource-checked

Prove that $0 \leq \ln(1+\alpha) \leq \alpha$ for all $\alpha \geq 0$

Source: Lean-Workbook row lean_workbook_plus_13953 (Apache-2.0). The unchanged proposition comes from the original record; its missing mathematical imports and namespaces are restored here.

Preamble

import Mathlib.Analysis.SpecialFunctions.Log.Basic
open Real

Formal statement

theorem WorkbookRestored.plus_13953 (α : ℝ) (hα : 0 ≤ α) : 0 ≤ Real.log (1 + α) ∧ Real.log (1 + α) ≤ α   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_13953; immutable original Prove2Me node 9427c2fe-9bae-49e8-a33f-e1118fe9d406

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