We consider two sequences of real numbers
![x_{1} \geq x_{2} \geq \ldots \geq x_{n}](/media/m/4/c/3/4c39387701faed90203a8295939d892a.png)
and
![\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.](/media/m/3/0/e/30e4365f741c9a001ff24973743b65e1.png)
Let
![z_{1}, z_{2}, .\ldots, z_{n}](/media/m/c/4/d/c4d99a7335c32d19fe8f9c1a10f11288.png)
be a permutation of the numbers
![y_{1}, y_{2}, \ldots, y_{n}.](/media/m/1/8/8/188d48141f4c01f339bebbd42e56bcd9.png)
Prove that
%V0
We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$