SCHWEITZER 2016 Problem 4

Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that $$a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)}$$ for all integers $m,n\ge 1$ , and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line.

Remark. A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$ , then $a(n)/n$ converges or tends to $(-\infty)$ .