intersection of infinite collection of finite sets? I know that there are questions asking like "intersection of a infinite collection of sets" and I can understand that the answer for that one is a null set, but I got a question here, in which all sets are finite and nonempty. Please take a look at the pic below.
It's a True/False question.
What I'm confused about is that the "infinity" sign in the interception. Since all sets are finite, does it mean that there are many sets in the chain are the same? (because the symbol used is for subsets not for proper subsets). Please give me some hints how I should think about this question.
Thank you

 A: Yes; it actually means that, there is some $N$ such that, for all $n>N$ we have $A_{N}=A_{n}$ - that is, all the sets are the same. This is basically what you're thought that $A_1$ has only finitely many subsets amounts to. The proof of this is basically the same as saying that the sequence of cardinalities $$|A_1|\geq |A_2|\geq |A_3|\geq \ldots$$
will be constant for large enough $N$ - that is, you cannot have an infinite descending sequence of natural numbers. (This is provable by induction, and essentially the idea induction captures in the first place)
Then, you can safely say that, for that $N$, we have $A_{i}\supseteq A_N$ for all $i$. Thus, the intersection of all of the terms of the sequence (which is what the $\infty$ in the intersection is trying to say) must contain $A_N$, since all the terms in the intersection do. However, it is also contained in $A_N$, since $A_N$ is a member of the set over which the intersection is being taken. Thus, it equals $A_N$, which is a finite, non-empty set.
