TITLE: What exactly is a countable union?
QUESTION [12 upvotes]: For example, $A_i$ be the set of all numbers divisible by $i$, consider:
$\bigcup_{n \in \mathbb{N}} A_n$
Is this union a countable union because I am enumerating over a countable set (the natural numbers)?

REPLY [16 votes]: Yes. Countable union means the index set over which one is taking the union is countable. In this case it is $\mathbb{N}$, which is countable. 
Incidentally, in this case, each of the $A_{i}$'s also happen to be countable, so what you have is a countable union of countable sets.