In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if and then Similarly, if and then Proof Consider the sum The two sequences are non-increasing, therefore and have the same sign for any . Hence . Opening the brackets, we deduce: hence An alternative proof is simply obtained with the rearrangement inequality, writing that Continuous version There is also a continuous version of Chebyshev's sum inequality: If f and g are real-valued, integrable functions over [a, b], both non-increasing or both non-decreasing, then with the inequality reversed if one is non-increasing and the other is non-decreasing. See also Hardy–Littlewood inequality Rearrangement inequality Notes Inequalities Sequences and series