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.

>>35

Why do you think it does??  You're looking at the (countable) set of all infinite sequences of coinflips which are eventually always zero, and just claiming that AC lets you pick a random element of that set, which is basically the same problem we started with.

http://en.wikipedia.org/wiki/Axiom_of_choice