A weighted shifted ratio sum has an upper bound in the total

Proved

WorkbookSource.base_52052

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,b,c>0$ . Prove that $\frac{a}{1+a}+\frac{2b}{2+b}+\frac{3c}{3+c}\le \frac{6(a+b+c)}{6+a+b+c}$

Source: InternLM Lean-Workbook, record lean_workbook_52052 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_52052 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a / (1 + a) + 2 * b / (2 + b) + 3 * c / (3 + c)) ≤ 6 * (a + b + c) / (6 + a + b + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_52052; Apache-2.0

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