autre
- 116
- 0
Homework Statement
\{x_{n}\}\in\mathbb{R^{+}} is a bounded sequence and r=\lim\sup_{n\rightarrow\infty}x_{n}. Show that \forall\epsilon>0,\exists finitely many x_{n}>r+\epsilon and infinitely many x_{n}<r+\epsilon.
The Attempt at a Solution
By definition of limit superior, r\in\mathbb{R} is such that \forall\epsilon>0, \exists N_{\epsilon} s.t. x_{n}<r+\epsilon, \forall n>N_{\epsilon}. This would imply that any x>r+\epsilon/is an upper bound on \{x_{n}\}. How do I show that there are finitely many such upper bounds? Is it because \{x_{n}\} is a bounded sequence that there must only be finite x_{n}&amp;gt;r+\epsilon ?