An alternative proof is simply obtained with the rearrangement inequality.

Note that the rearrangement inequality makes no assumptions on the signs of the real numbers.

It can also be proven using the rearrangement inequality.

This is a consequence of the rearrangement inequality.

By the rearrangement inequality, we have , so the fraction on the lesser side must be at least .

This proof assumes knowledge of the rearrangement inequality and the arithmetic-geometric mean inequality.

Note that this result generalizes the rearrangement inequality and Chebyshev's sum inequality.

As we have used the rearrangement inequality and the arithmetic-geometric mean inequality, equality only occurs when and the triangle is equilateral.

Chebyshev's sum inequality, rearrangement inequality (These two articles may shed light on the mathematical properties of Spearman's ρ.)

Ephremidze The Rearrangement Inequality for the Ergodic Maximal Function Georgian Math.

Many famous inequalities can be proved by the rearrangement inequality, such as the arithmetic mean - geometric mean inequality, the Cauchy-Schwarz inequality, and Chebyshev's sum inequality.

The scalar product of the two sequences is maximum because of the rearrangement inequality if they are arranged the same way, call and the vector shifted by one and by two, we have: