The Limit Superior and Bounded Sequences

[autre]

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}>r+\epsilon ?

 

[Office_Shredder]

How do I show that there are finitely many such upper bound

I'm not sure what you mean here, but what you've written basically answers the first part of the question. If for n>N_{\epsilon}, x_n<r+\epsilon, then there are at most N_{\epsilon} values of x_n which are larger than r+\epsilon

 

[autre]

I'm not sure what you mean here, but what you've written basically answers the first part of the question. If for n>N_{\epsilon}, x_n<r+\epsilon, then there are at most N_{\epsilon} values of x_n which are larger than r+\epsilon

I meant that "there are finitely many such x\in(x_{n})". How do I get started proving that there are infinitely many x\in(x_{n}) s.t. x<r+\epsilon?

 

[poochie_d]

How do I get started proving that there are infinitely many x\in(x_{n}) s.t. x<r+\epsilon?

Suppose there were only finitely many x_n such that x_n < r + \epsilon. Would this in any way contradict the given facts? (Think about the definition of lim sup)