Lean-Workbook Plus 61757: Exponential inequality

Proved

WorkbookRestored.plus_61757

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

lean-workbookrestored-importssource-checked

For $-1<x<0$ , $e^x<1<1/(1+x)$ ; for $x>0$ , $e^x>1>1/(1+x)$ .

Source: Lean-Workbook row lean_workbook_plus_61757 (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_61757 : ∀ x ∈ Set.Ioo (-1 : ℝ) 0, exp x < 1 ∧ 1 < 1 / (1 + x) ∧ ∀ x ∈ Set.Ioi 0, exp x > 1 ∧ 1 > 1 / (1 + x)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_61757; immutable original Prove2Me node 4c08e189-022b-4c66-8aa6-54b181129d98

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