Actually, a countable union of countable sets is countable

Technical note: strictly this requires the Axiom of Choice, or at least some weaker version of it. For each of your countable sets, there is at least one way of counting it; but to count the whole lot you need to pick one way of counting each set. This is exactly the kind of thing that can fail to happen without Choice. You don't need "very much" Choice; e.g., the axiom of choice for countable collections is enough; but it turns out that the axiom of choice for countable collections of countable sets is not enough.