upper limit of a sequence of sets Let $\langle A_n:n\in\Bbb N\rangle$ be a sequence in $X$,$\lim\sup A_n=\bigcap_{n\in N}\bigcup_{k>n} A_k,$,can we conclude that $A_n\subset lim\sup A_n$ when n tends to $\infty$? If not,what,s the relationship between $A_n$ and $lim\sup A_n$ when $n$ is large enough.

Put $A_{2n}:= \{0,n\}$ and $A_{2n+1}:=\{n\}$. Then $\limsup A_n = \{0\}$ but there is no relation between $A_n$ and $\limsup A_n$ in the sense of inclusion. In $\limsup A_n$ you get exactly those elements which are contained in infinitely many $A_n$'s so you have enough freedom to destroy any strict relation in this direction.
