User blog comment:Edwin Shade/An Injection From The Countable Ordinals To The Reals/@comment-1605058-20171219180546/@comment-1605058-20171219220143

You can't really define the fundamental sequences using the axiom of choice. What it allows you to do is to pick, for every countable limit ordinal, one out of all the possible fundamental sequences, and construct an assignment \((\alpha,n)\mapsto\alpha[n]\) for all those ordinals at the same time.