Why does $\bigcap_{m = 1}^\infty ( \bigcup_{n = m}^\infty A_n)$ mean the limit superior of sequence of set?
I'm not getting it. ${A_n}$ is a sequence of set in $S$. I do know what limsup means for a bounded sequence; however I am totally baffled at the sequence of set. Can anyone help?
Consider the statement $x \in \bigcap_{m=1}^\infty \bigcup_{n=m}^\infty A_n$. This means exactly that $x \in \bigcup_{n=m}^\infty A_n$ for every $m$. (Here I have expanded the definition of the intersection.) This means exactly that for every $m$, $x \in A_n$ for some $n \geq m$. (Here I have expanded the definition of the union.) This means exactly that $x$ is in infinitely many of the $A_n$.