Let $x_{n} \in \mathbb{R}$ for all $n \in \mathbb{N}$. It is clear that $x_{n} \leq 0$ for large $n$ (i.e. there is some $N$ such that $x_{n} \leq 0$ for all $n \geq N$) implies $\limsup_{n \to \infty}x_{n} \leq 0$. Since it is convenient to have a shorthand for a phrase of the form "$x_{n} \leq 0$ eventually in $n$ (or for large $n$)", it is tempting to exploit the $\limsup$. Unfortunately, this seems possible if and only if we are dealing with a strict inequality. For a generic inequality, the equality case admits some careful treatment as the sequence $(1/n)$ satisfies $\limsup_{n \to \infty}x_{n} \leq 0$ but there is certainly no $n$ such that $x_{n} \leq 0$.
Now a natural question is under what condition(s) we have the converse? A condition less restrictive in that, for instance, it would not be as long as what is to be shortened, would be a priority.
For the sake of writing something different (though still equivalent unless $x_n=0$ infinitely often)