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.

Flip a coin forever so that heads is 1 and tails is 0.  You will generate your number in binary.  You won't ever finish, but that's ok because you math weirdos defined real numbers as limits of inifinite sums anyway.