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.

>>9 here...

>>10
You can't do the coin flip thing, since you're not guaranteed to ever get a finite number.  At some point, you'd have to have chosen a finite number of digits, and get nothing but zeros from then on.

Your second statement is correct here; after a certain point, you'd need an infinite number of zeros. This is not a problem. This is in fact what the axiom of choice allows us to do.

>>17
>>18
What does this have to do about convergence?