According to the Set Theory book I am studying, the Axiom of Choice is required to prove that the union of a countable number of at most countable sets is itself at most countable. I seem to have proven this without using choice but I am certain that I made a mistake somewhere! As I use a lot of custom macros, I posted the proof here. I hope this isn't breaking any rules.
I'm pretty sure the error is in Lemma 1 and something that I am doing assumes/requires choice (or well-ordering) but I do not know what. Any help in pointing out the problem would be much appreciated!