that is, is it true that we can not prove the existence of function such that without invocation of Axiom of Choice, where is the set of all natural numbers and is countably infinite?
Well, I understand: to simply prove the existence of such an index does not need Axiom of Choice, but singling out one such index out of a set of indices needs Axiom of Choice. So the first sentence "Let " of the proof of Th2.25 of Apostol's "Mathematical Analysis" is actually using Axiom of Choice implicitly.