lean_workbook_plus_54388

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Suppose that $f(x)$ and $g(x)$ are positive for all $x$ . If $f(x)$ and $g(x)$ are both monotonically increasing, then must $f(x)\cdot g(x)$ be monotonically increasing?

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_54388  (f g : ℝ → ℝ)
  (hf : ∀ x, 0 < f x)
  (hg : ∀ x, 0 < g x)
  (hf' : ∀ x y, x ≤ y → f x ≤ f y)
  (hg' : ∀ x y, x ≤ y → g x ≤ g y)
  : ∀ x y, x ≤ y → f x * g x ≤ f y * g y   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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