SELECTION and TRAINING SESSION

$$f(x)=\{\begin{array}{ll}x+\frac{1}{2},& \text{if }x<\frac{1}{2},\\ x^2,& \text{if }x\ge \frac{1}{2}.\end{array}$$ 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 $$(a_n-a_{n-1})(b_n-b_{n-1})<0.$$