# Apostol Proof for Cantor Intersection Theorem

I am trying to understand the proof for the following theorem from Apostol:

Here is the proof:

I don't understand the parts underlined in red. For the first part, why is it trivial if each $$Q$$ is finite? For the second part, how does $$Q$$ contain all points but possibly a finite amount of points of $$A$$? Does it have something to do with the first condition $$Q_{k+1}\subseteq Q_{k}$$?

1. Suppose that $$Q_i$$ only has $$n$$ elements. Since $$Q_i\supset Q_{i+1}\supset Q_{i+2}\supset\cdots$$, you have$$n=\#Q_i\geqslant\#Q_{i+1}\geqslant\#Q_{i+2}\geqslant\cdots$$So, you have a decreasing sequence of elements of $$\{1,2,\ldots,n\}$$. But then the sequence $$\#Q_i,\#Q_{i+1},\#Q_{i+2},\ldots$$ must becaom stable after some point. So, for some $$N\geqslant i$$, you have $$Q_N=Q_{N+1}=Q_{N+2}=\cdots$$ and therefore $$\bigcap_{n\in\mathbb N}Q_n=Q_N\neq\emptyset$$.
2. Yes, it has to do with that. Note that $$x_1$$ belongs to every $$Q_n$$, that $$x_2$$ belongs to every $$Q_n$$ except perhaps for $$Q_1$$, then $$x_3$$ belongs to every $$Q_n$$ except perhaps for $$Q_1$$ and $$Q_2$$ and so on. So, for each $$Q_n$$, all but possibly a finite number of elements of $$A$$ belong to $$Q_n$$.

If at least one $$Q$$ is finite, say $$|Q_n|=N$$, then $$|Q_k|$$, $$k\ge N$$, is a non-decreasing sequence of positive integers. It follows that $$|Q_k|$$ is constant for $$k$$ large enough. But then $$Q_k$$ is also constant for these large $$k$$.

By the nesting property, $$x_j\in Q_k$$ for all $$j\ge k$$.