Suppose there is a sequence $\{x_n\}_{n\in\Bbb N}$ s.t. $$0\le x_{n+m} \le x_n\cdot x_m\quad\forall m,n\in\Bbb N$$ show that $$\lim_{n\to\infty}\left(x_n\right)^{1/n}=\xi\in\Bbb R$$
Seeing the "$\forall n,m\in\Bbb N$", the first thing to pop into my mind is Cauchy's criterion for sequential convergence, but this approach has got me nowhere, because there seems to be nothing I can do to control $|(x_n)^{1/n}-(x_m)^{1/m}|$.
Another thing to be noticed is that in my text book this exercise belongs to the "infinite sum" category instead of "sequence", thus it is my notion that maybe there is a very tricky method to connect this sequence to a certain infinite sum and finish the proof?
Any help or hint will be appreciated. Best regards!
