$$f(x) = \begin{cases} x + \frac{1}{2}, & \text{if } x < \frac{1}{2}, \\ x^2, & \text{if } x \ge \frac{1}{2}. \end{cases}$$
Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and
$$a_n = f(a_{n-1}),\ b_n = f(b_{n-1}) \text{ for } n > 0.$$
Show that there exists a positive integer $n$ such that
$$(a_n - a_{n-1})(b_n - b_{n-1}) < 0.$$