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Let $x_n$ be a bounded sequence of real numbers, let $x_*$=lim inf$(x_n)$ and $x^*$=lim sup$(x_n)$. Also, let e > 0. Prove that there exists an N in the naturals such that $x_n$$]x_*-e,x^*+e[$ for all n>N. I wanted to go by contradiction. So, finding a subsequence that is never in the interval. But how would I do that?

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Assume there exists an $\epsilon >0$ for which you have for every $N \in \mathbb{N}$ at least one $n>N$ such that $x_n > x^* + \epsilon$.

Now recall the definition: $$\limsup\limits_{n\to\infty} x_n := \inf\limits_{N\in\mathbb{N}} \sup\limits_{n \geq N} x_k$$

What can you say about $\sup\limits_{n \geq N} x_n$?

Note: if you need more just say it

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