Convergence of $\sqrt[n]{s_n}$ given $s_ns_m \leq s_{n+m}$

Question

Given $(s_n)_{n \geq 1}$, a sequence of positive integers and the inequality $s_ns_m \leq s_{n+m}$ for all integers $n, m$, how does one go about proving that the sequence $(\sqrt[n]{s_n})_{n \geq 1}$ converges?

It is quite easy to see that the sequence is monotone increasing, so all we need is an upper bound for all of its terms. I don't quite see how to do this, since the inequality only seems to give a lower bound..

Answer 1

Consider the sequence $s_n = 2^{n^2}$ and observe that \begin{align} s_ns_m = 2^{n^2+m^2} \leq 2^{(m+n)^2}= s_{n+m} \end{align} but we see that \begin{align} \sqrt[n]{s_n} = 2^n\rightarrow \infty. \end{align}