What is the weakest assumption to be satisfied so that $\lim \sup x_n \in \mathbb R$. The same question for $\lim \inf x_n$.
Note that $\mathbb R$ does not include $\pm \infty$.
Should $x_n$ necessarily be bounded from above and below, or a weaker assumption will do?
The correct condition is that $\limsup x_n$ is finite iff $x_n$ is bounded from above and has a subsequence that's bounded from below.