I am reading a paper that contains the following limit:
$$\lim_{n \to \infty}\frac{log(a_n)}{n}$$
where we have the following growth control on $a_n$:
$a_{n+m} \leq a_n a_m$.
I am trying to prove that the above limit exists, using this fact (according to the paper it is supposed to be easy, but I'm not seeing it). I was also told that the fact that this limit exists is a standard argument, but I can't find any material on sequences that grow sub-exponentially.
I was hoping that I could get some hints or references. Thank you.
