set_limes_superior

Set limes superior

context	$A\in \text{Seq}(X)$
definition	$\underset{n\to\infty}{\limsup}A_n\equiv{\bigcup_{n=1}^\infty}\left({\bigcap_{k=n}^\infty}A_k\right)$

We have that

$\underset{n\to\infty}{\limsup}A_n\subseteq \underset{n\to\infty}{\liminf}A_n,$

$\underset{n\to\infty}{\limsup}A_n=\underset{n\to\infty}{\liminf}A_n,$

then we call it

$\underset{n\to\infty}{\lim}A_n$

and say $A$ is convergent.

For a more possibly more elucidating explaination, see my answer in this Math.Se thread this Math.SE thread

Parameterized set