Question 152
Define a function $f : (0,1) \to (0,1)$ as follows:
$$
f(x) = \begin{cases} x + \frac{1}{2} & \text{if } x \lt \frac{1}{2}, \\ x^2 & \text{if } x \ge \frac{1}{2}. \end{cases}
$$
Let $a, b$ be two real numbers with $0 \lt a \lt b \lt 1$. Define two sequences as follows:
$a_0 = a$, $b_0 = b$, and $a_n = f(a_{n-1})$, $b_n = f(b_{n-1})$, $n = 1, 2, \dots$
Prove that there exists a positive integer $n$ such that
$(a_n - a_{n-1})(b_n - b_{n-1}) \lt 0.$
$$
f(x) = \begin{cases} x + \frac{1}{2} & \text{if } x \lt \frac{1}{2}, \\ x^2 & \text{if } x \ge \frac{1}{2}. \end{cases}
$$
Let $a, b$ be two real numbers with $0 \lt a \lt b \lt 1$. Define two sequences as follows:
$a_0 = a$, $b_0 = b$, and $a_n = f(a_{n-1})$, $b_n = f(b_{n-1})$, $n = 1, 2, \dots$
Prove that there exists a positive integer $n$ such that
$(a_n - a_{n-1})(b_n - b_{n-1}) \lt 0.$