Sequence such that $x_{n+m} \leq x_n+x_m$

Question

I have a couple of questions about the following type of sequences :

Let $(x_n)$ be a sequence such that $x_n \geq 0$ $\forall n \in \mathbb N$ and that $$x_{n+m} \leq x_n+x_m \quad \forall n,m \in \mathbb N $$

1. In order to show that the sequence $(\frac{x_n}{n})$ converges, all proofs "guess" the limit that is $\inf(\frac{x_n}{n})$ and then use Euclidean division. What if I cannot guess it by myself (which actually happened) and do not think of using Euclidean division ?
2. What happens if the sequence's property is $$x_{n+m} \geq x_n+x_m \quad \forall n,m \in \mathbb N $$ I feel like that here the sequence $(\frac{n}{x_n})$ will converge (tried with $x_n=e^n$), but I am not really sure...