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.

>>55
Get it right, it's the limit of the sequence, not the sum of the sequence.
.1, .5, .9, .99, .999, ... and
.9, 1.01, .999, 1.0001, .99999, ...
are very different sequences with the same limit.  It seems rather nonsensical to define real numbers as sequences, as you seem to be suggesting.