which of these is correct:
Let $A_n$ and $B_n$, $n\in\Bbb N$ be nonempty subsets of $\Bbb R$ such that $A_1\supseteq A_2\supseteq A_3\supseteq\dots$ and $B_1\subseteq B_2\subseteq B_3\subseteq\dots$. Let cardinality of $A_n$ be $a_n$ and the cardinality of $B_n$ be $b_n$.
Then the cardinality of $\cap_{n=1}^\infty A_n$ is $\lim\limits_{n\to \infty }a_n$,
Then the cardinality of $\cup_{n=1}^\infty B_n$ is $\lim\limits_{n\to \infty }b_n$.
Which of 1. and 2. is correct?
I thought about finite cardinalities, because I have only once heard of 'biggest set with countable number of elements'.
Let $B_n=\{1,2.\dots,n\}$, so, $\lim\limits_{n\to \infty }b_n=\infty$, but $\infty$ is not a cardinality. Is this correct way to approch?
1 is false; think about $$A_n = (0,1/n).$$
2 is should be true, for an appropriately chosen topology on the set of cardinal numbers between $0$ and $\beth_1$. But you have to choose the topology first, which is what Asaf's comment is getting at.
Here's an explanation of the $\beth_\alpha$ notation; in this context, I'm using $\beth_1$ because it equals the cardinality of the continuum.