Let
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
be an integer greater than
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
. Define
![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{for }i =...](/media/m/8/0/2/80297c16d4034850b8005614863d0012.png)
where
![[z]](/media/m/1/b/e/1be90bc55017944743736332d25924af.png)
denotes the largest integer less than or equal to
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
. Prove that
%V0
Let $n$ be an integer greater than $1$. Define
$$x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i =...$$
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
$$\min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]$$