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.

I know a way you can finish! Take an infinite number of robots that can each flip one 10 sided die. They all flip at once and display their digit. Done.