But what does it mean to pick an element infinitely many times? You have to be very careful about your assumptions there, an inductive proof would only prove you can pick an arbitrarily large number of such elements. To say that you can make a set picking one element each from a infinite collection of sets, even countably infinite, is something you cannot prove inductively. It requires another axiom.

And admitting this axiom suddenly paradoxes like Banach-Tarski are possible.