There is a theorem saying that:
$\limsup x_n = \sup{z_k}$ where ${z_k}$ is a set of limit points for the sequence ${x_n}$.
All sets and sequences are real.
The definition of limit superior is: $$\limsup x_n = \lim_{n \to \infty} \sup_{m \ge n}\{x_m\}=\inf_{n} \sup_{m \ge n}\{x_m\}$$
Limit point for a sequence is a point such that in any neighborhood of it there is a point from the sequence other than the original point itself.
If $\lim \sup x_n$ is finite, then clear enough it is a limit point itself, and there might be no other limit point which exceed $\lim \sup x_n$.
But how to proceed with the proof if $\lim \sup x_n= \pm \infty$?