I think if you're trying to "prove" this using axioms, you've already lost. The problem isn't that you can't come up with axioms to convince people you have a

I think if you're trying to "prove" this using axioms, you've already lost. The problem isn't that you can't come up with axioms to convince people you have a proof - the problem is with people not understanding that 0.99999.... is not a number - it's one representation of an abstract entity called a number. The problem is, the maths required to actually define the concept of a number is fairly complicated, so it's hard to explain to someone why all of these axioms make sense in the first place.