Question 27
Let $c \gt 2$, and let $a(1), a(2), \ldots$ be a sequence of nonnegative real numbers such that
$$
a(m+n) \leq 2 a(m)+2 a(n) \quad \text{ for all } m, n \geq 1, \tag{1}
$$
and
$$
a\left(2^{k}\right) \leq \frac{1}{(k+1)^{c}} \quad \text{ for all } k \geq 0 \tag{2}
$$
Prove that the sequence $a(n)$ is bounded.
$$
a(m+n) \leq 2 a(m)+2 a(n) \quad \text{ for all } m, n \geq 1, \tag{1}
$$
and
$$
a\left(2^{k}\right) \leq \frac{1}{(k+1)^{c}} \quad \text{ for all } k \geq 0 \tag{2}
$$
Prove that the sequence $a(n)$ is bounded.