Školjka

%V0 Let $c > 2,$ and let $a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that $$a(m + n) \leq 2 \cdot a(m) + 2 \cdot a(n) \text{ for all } m,n \geq 1,$$ and $a\left(2^k \right) \leq \frac {1}{(k + 1)^c} \text{ for all } k \geq 0.$ Prove that the sequence $a(n)$ is bounded. Author: Vjekoslav Kovač, Croatia