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 [itex]x>r+\epsilon/[itex] 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)