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Homework Statement
[itex]\{x_{n}\}\in\mathbb{R^{+}}[/itex] is a bounded sequence and [itex]r=\lim\sup_{n\rightarrow\infty}x_{n}[/itex]. Show that [itex]\forall\epsilon>0,\exists[/itex] finitely many x_{n}>r+\epsilon and infinitely many [itex]x_{n}<r+\epsilon[/itex].
The Attempt at a Solution
By definition of limit superior, [itex]r\in\mathbb{R}[/itex] is such that [itex]\forall\epsilon>0[/itex], [itex]\exists N_{\epsilon}[/itex] s.t. [itex]x_{n}<r+\epsilon, \forall n>N_{\epsilon}[/itex]. This would imply that any [itex]x>r+\epsilon/[itex] is an upper bound on [itex]\{x_{n}\}[/itex]. How do I show that there are finitely many such upper bounds? Is it because [itex]\{x_{n}\}[/itex] is a bounded sequence that there must only be finite [itex]x_{n}>r+\epsilon[/itex] ?