PL-OM 2025 Problem 6

A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy $$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\; b_{n+1}=b_n+f(a_n)$$ for every $n\geq 1$ . Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$ .