Lean-Workbook Plus 45623: Exponential inequality

Proved

WorkbookRestored.plus_45623

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

lean-workbookrestored-importssource-checked

For a positive natural number $n$ and $0\le x\le n$ , $(1-x/n)^n\le e^{-x}$ .

Source: Lean-Workbook row lean_workbook_plus_45623 (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
import Mathlib.Analysis.SpecialFunctions.Pow.Real
open Real

Formal statement

theorem WorkbookRestored.plus_45623 (x : ℝ) (n : ℕ) (hn : 0 < n) (hx : 0 ≤ x ∧ x ≤ n) :
  (1 - x / n)^n ≤ exp (- x)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_45623; immutable original Prove2Me node 4d93f5a9-54f5-44f2-a084-a990812f3296

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