In Bartle's Introduction to Real Analysis, he defines the limit superior $x^*$ of a bounded sequence of reals $(x_n)$ as the infimum of the set $V$ of $v \in \mathbb{R}$ such that $v < x_n$ for at most a finite number of $n \in \mathbb{N}$. He then shows that if $x^* = \limsup x_n$ and $\epsilon > 0$, then there are at most a finite number of $n \in \mathbb{N}$ such that $x^* + \epsilon < x_n$ and an infinite number of $n \in \mathbb{N}$ such that $x^* - \epsilon < x_n$.
I had a question regarding the proof of the fact in the last sentence. Given $\epsilon > 0$, since $x^* = \inf V$ we have that $\exists v \in V$ such that $x^* \leq v < x^* + \epsilon$. Bartle then claims that this implies $x^* \in V$, though I am not sure why. Here is the proof taken directly.
