Limsup, liminf simple question

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SUMMARY

The discussion clarifies the difference between the notations \limsup\limits_{n\in\mathbb{N}} x_n and \limsup\limits_{n\to\infty} x_n. The latter notation is the correct form to express the limit superior as it approaches infinity, while the former lacks meaning as a limit. The correct usage of \sup_{n\in\mathbb{N}} is also highlighted, indicating that it pertains to the supremum of a set rather than a limit.

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St41n
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Is [tex]\limsup\limits_{n\in\mathbb{N}} x_n[/tex] any different than [tex]\limsup\limits_{n\to\infty} x_n[/tex] ?
 
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The limsup is a limit, so the notation on the left doesn't really have any meaning and the notation on the right is correct. You use something like what you have on the left for just the sup, i.e. [tex]\sup_{n\in\mathbb{N}}[/tex] of some set.
 
that makes sense, thanks
 

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