ILL 1977 Problem 28

Let $n$ be an integer greater than $1$. Define

$$x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \text{for }i = 1, 2, \ldots\ ,$$

where $[z]$ denotes the largest integer less than or equal to $z$. Prove that $$\min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]$$