# Homework Help: Limit superior question

1. Jan 21, 2012

### autre

1. The problem statement, all variables and given/known data

$\{x_{n}\}\in\mathbb{R^{+}}$ 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 $x_{n}<r+\epsilon$.

3. 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 $x>r+\epsilon/[itex] is an upper bound on [itex]\{x_{n}\}$. 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 $x_{n}>r+\epsilon$ ?

2. Jan 21, 2012

### Office_Shredder

Staff Emeritus
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 xn which are larger than $r+\epsilon$

3. Jan 22, 2012

### autre

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. $x<r+\epsilon$?

4. Jan 23, 2012

### poochie_d

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)