Sea $n$ un entero mayor que $1$. Definimos \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{para }i = 1, 2, \ldots\ ,\] donde $[z]$ denota el mayor entero menor o igual a $z$. Demuestre que \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]