KöMaL 2019 A741

Let $f$ be a function defined on the positive integers with $f(n)\ge 0$ and $f(n)\le f(n+1)$ for all $n$. Prove that if $\sum\limits_{n=1}^\infty \frac{f(n)}{n^2}$ diverges, there exists a sequence $a_1,a_2,\dots$ such that the sequence $\frac{a_n}{n}$ hits every rational number, while $a_{n+m}\le a_n+a_m+f(n+m)$ holds for every pair $n$, $m$.