Lean-Workbook Plus 18746: Exponential inequality

Proved

WorkbookRestored.plus_18746

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

lean-workbookrestored-importssource-checked

Prove that $\frac{u}{1+e^{-u}}<u$ for all $u>0$ .

Source: Lean-Workbook row lean_workbook_plus_18746 (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_18746 (u : ℝ) (hu : 0 < u) : u / (1 + exp (-u)) < u   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_18746; immutable original Prove2Me node 4e16dec5-12ef-4f50-a897-356aac6d5f35

View graph