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How is closed under finite union not the same as closed under countable union?

I know that closed under countable union should be a stronger requirement than closed under finite union. So why does the following argument not work?

Let {$A_1,A_2,...$} be a countably infinite sequence of sets in F. Then if F is closed under countable union, the countable union of {$A_1,A_2,...$} should also be in F.

However, If we assume F was closed only under finite union, couldn't you use induction to show that the countable union of {$A_1,A_2,...$} will also be in F? That is $A_1 \bigcup A_2$ is in F, $A_1 \bigcup A_2 \bigcup A_3$ in in F and so on..

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The standard counterexample goes something like this.

Take $A_n = \{0,1,\dots,n\}$, and let $S$ be the collections of all sets $A_n$. Then we have that finite unions, say $\bigcup_k A_{n_k}$, that this belongs to $S$, just by considering that the $A_{\max {n_k}}$ will be a element of $S$.

But consider the countable union $\bigcup_{n \in \mathbb{N}} A_n = \{1, 2, \dots\}$. This is not an element of $S$, which can be shown by a simple argument from contradiction, and follows from the fact that this is no longer a finite set.

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