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Can someone give an elegant proof to the following question?

For any sequence, there exist a monotonic subsequence whose limit is $\limsup S_n$ and there exists a monotonic subsequence whose limit is $\liminf S_n$.

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    You're probably talking about sequences of real numbers. (You need to work with some set which has an ordering, so that the notions of limit superior and limit inferior make sense.) You should specify this in your question.2011-08-27

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Since $S:=\limsup x_n$ is a cluster point of the real sequence $x=(x_n)$, there exists a subsequence of $x$ which converges to $S$.

Using the fact that every real sequence has a monotone subsequence you obtain the wanted result.

You can find proofs of both results here (and probably in many other texts).

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    good,thank you,and that's useful2011-08-27