Union of the Intersection of Infinite Sets So, I'm trying to make sense of this question:

"Suppose we have an $A_n$ for each n $\in$ N (i.e. N is the Naturals. Each $A_n$ is an arbitrary set). Fill in the blanks with four words to get a true statement. Justify your answer.

x $\in$ $\bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}$ $A_n$ iff x $\in$ $A_n$ for [blank] [blank] [blank] [blank] n $\in$ N"

So far, I understand that for a given k, if x $\in$ ($\bigcap_{n=k}^{infinity}$ $A_n$ = $A_k$ $\cap$ $A_{k+1}$ $\cap$ $A_{k+2}$ $\cap$ ....) $\implies$ x $\in$ $A_n$ for all n$ 
\geqslant$ k

But I'm stuck on interpreting the next stage. If x $\in$ $\bigcup_{k=1}^{\infty}\bigcap_{n=k}^{\infty}$ $A_n$, then x $\in$ ({$A_1$ $\cap$ $A_{2}$ $\cap$ $A_{3}$ $\cap$...} $\bigcup$ {$A_2$ $\cap$ $A_{3}$ $\cap$ $A_{4}$ $\cap$...} $\bigcup$....). Which just means that $\exists$ some b $\in$ N s.t. x $\in$ $A_n$ for all $n\geqslant$ b. 

Have I interpreted this all correctly, and if so how could I translate what I've got so far into the 4 words? If I've misinterpreted the situation, have you any thoughts as to what I'm lacking?

Many thanks, really appreciate it.
"all except finitely many" is what I would write there. That is the definition of [\liminf ]. By the way, remember the command for "infinite" is \infty.
"all except finitely many" is what I would write there. That is the definition of [\liminf ]. By the way, remember the command for "infinite" is \infty.