You should be able to solve this.

Create a method to choose a random integer between 0 and infinity such that no integer is more likely to be chosen than any other.

Yes it is possible, I don't care what your probability book says (Notice I didn't even use the word "probability").  You may assume the axiom of choice.

>>59
I suppose, but they're the same thing.  Two items are in the same equivalence class if they have the same limit, which is equivalent to the "definition" of real numbers I stated.
From a logic point of view, I can see how one would be more fundamental/easier to show from fundamental stuff than the other.
I take it you mean "cauchy sequences of rational numbers".