Infinitesimals are too small

I am getting to the following intuition as to why it's not appropriate to resolve zero-probability problems, like the problem of the probability of a particular number being picked in the case of an infinite lottery. The intuition is that the infinitesimals are too small to do the job: infinitesimals are too small to bridge the gap between zero and one, even if you have infinitely many of them. Wish I had time to go into why I am getting this intuition, but I need to grade assignments on mathematical induction instead.