ISL 2007 A5

Let $c>2$, and let $a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that $$a(m+n) \leq 2 a(m)+2 a(n) \text { for all } m, n \geq 1 \text {, }$$ 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. (Croatia)