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Theorem Every sequence {$s_n$} has a monotonic subsequence whose limit is equal to $\limsup s_n$. I think to show that there exist a monotonic subsequence is kind of straight forward but I could show that there exist such subsequences whose limit is $\limsup s_n$.

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    It is known that $$ \limsup\limits_{n\to\infty} s_n=\sup\{\lim\limits_{k\to\infty} s_{n_k}:\{s_{n_k}:k\in\mathbb{N}\}-\text{ convergent subsequence of }\{s_n:n\in\mathbb{N}\}\}\} $$ So you can easily extract such a subsequence2012-07-16
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    You probably mean real sequence (=sequence of real numbers), but perhaps it would be better to mention it explicitly in the question.2012-07-16

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