Let $(x_{n})\in\mathbb{R}^{+}$ be bounded and let $x_{0}=\lim\sup_{n\rightarrow\infty}x_{n}$. $\forall\epsilon>0$, prove that there are infinitely many elements less than $x_{0}+\epsilon$ and finitely many terms greater than $x_{0}+\epsilon$.
My attempt: By definition of limit superior, $\forall\epsilon>0$, $\exists N_{\epsilon}\in\mathbb{N}$ s.t. $\forall n>N_{\epsilon}$, $x_{n}
I think my proof is probably incomplete/too informal. What do you think?