I was wondering about this problem, and haven't been able to solve it at all. Let $(x_n)$ be a real non-negative sequence, such that $x_{n+m}\le x_mx_n$, for every $m,n\in\mathbb{N}$. I'm asked to prove $\sqrt[n]{x_n}$ converges.
I use $y_n=\frac{x_n}{x_1^n}$, and it has a limit $a$ because of monotony. If $a\ne 0$, I got to prove $\sqrt[n]{y_n}\rightarrow 1$, so $\sqrt[n]{x_n}\rightarrow x_1$, but I don't know what to do if $a=0$.
Thanks for your help.