(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[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].

3. 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] ?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Limit superior question

**Physics Forums | Science Articles, Homework Help, Discussion**