Let
![c > 2,](/media/m/6/8/3/6837fa2348d802e29471871f84004241.png)
and let
![a(1), a(2), \ldots](/media/m/8/1/1/8119f281391576ed73223016de09f535.png)
be a sequence of nonnegative real numbers such that
and
![a\left(2^k \right) \leq \frac {1}{(k + 1)^c} \text{ for all } k \geq 0.](/media/m/e/6/6/e665cdcd98e3a278862ca6dcc2c25043.png)
Prove that the sequence
![a(n)](/media/m/1/d/7/1d727182bc04bf615d2f9347d052cd83.png)
is bounded.
Author: Vjekoslav Kovač, Croatia
%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