Let xi,yix_i, y_ixi,yi (i=1,2,…,n)(i = 1, 2, \ldots, n)(i=1,2,…,n) be real numbers such that
x1≥x2≥⋯≥xn and y1≥y2≥⋯≥yn.x_1 \geq x_2 \geq \cdots \geq x_n \text{ and } y_1 \geq y_2 \geq \cdots \geq y_n.x1≥x2≥⋯≥xn and y1≥y2≥⋯≥yn.
Prove that, if z1,z2,⋯ ,znz_1, z_2, \cdots, z_nz1,z2,⋯,zn is any permutation of y1,y2,⋯ ,yny_1, y_2, \cdots, y_ny1,y2,⋯,yn, then
∑i=1n(xi−yi)2≤∑i=1n(xi−zi)2.\sum_{i=1}^{n}(x_i - y_i)^2 \leq \sum_{i=1}^{n}(x_i - z_i)^2.i=1∑n(xi−yi)2≤i=1∑n(xi−zi)2.