Limsup, liminf simple question

[St41n]

Is $$\limsup\limits_{n\in\mathbb{N}} x_n$$ any different than $$\limsup\limits_{n\to\infty} x_n$$ ?

 

[n!kofeyn]

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. $$\sup_{n\in\mathbb{N}}$$ of some set.

 

[St41n]

that makes sense, thanks