"Consider an experiment whose sample space consists of a countably infinite number of points. Show that not all points can be equally likely. Can all points have positive probability of occurring?"
The only way I can think of proving this is by contradiction, but I don't know if I believe my own proof:
If each point is equally likely then there is a 1/n chance of it occurring. However:
lim n->infinity of 1/n = 0
All the probabilities of the sample space added together should equal 1, but if the limit is 0 then it follows that 0+0+0+... = 0 is not equal to 1.
I think the problem with this proof is that 1/n does not really equal 0...it equals epsilon. And epsilon*n should be 1. Is this proof completely off and is there a better way?