Question 11
(NET 3) Let $n$ be an integer greater than $1$. Define
$$x_1=n,\quad y_1=1,\quad x_{i+1}=\left[\frac{x_i+y_i}{2}\right],\quad y_{i+1}=\left[\frac n{x_{i+1}}\right]\qquad i=1,2,\ldots,$$
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
$$\min\{x_1,x_2,\ldots,x_n\}=[\sqrt n].$$
$$x_1=n,\quad y_1=1,\quad x_{i+1}=\left[\frac{x_i+y_i}{2}\right],\quad y_{i+1}=\left[\frac n{x_{i+1}}\right]\qquad i=1,2,\ldots,$$
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
$$\min\{x_1,x_2,\ldots,x_n\}=[\sqrt n].$$