Quote by jgens
In the real number system, there are no infinitesimal elements. The same is true in the extended reals and projective reals as well. In fact, most mathematicians rarely (if ever) do any work that uses formal infinitesimals. There are number systems that have infinitesimal elements (like the hyperreal numbers), but most of these have roots in model theory and are fairly difficult to define formally. If you are interested, nonstandard analysis is the subject that deals with the calculus of these infinitesimal numbers, but nonstandard analysis is far from one of the more active areas of research in analysis.
Therefore, it is often best not to resort with reasoning using infinitesimals. Without using their formal properties, it is easy for your intuition to deceive you. It turns out most people have terrible intuition when it comes to infinitesimals.
Now, it is important to note that [itex]lim_{h \to 0} h = 0[/itex]; that is, the value of the limit is 0. The limit is not infinitesimally close to 0, but actually is 0. This is an extremely important point to understand.
Finally, keeping what I've said above in mind, something with probability 0 can occur in just the same manner as something with probability 1 not occurring.

Hey, thanks for your patience. You haven't really argued on how something with probability 0 can occur. I'm not completely convinced at the moment. I'll try to put it in words for the sake of argument; to me 0 is absolutely nothing, so for absolutely nothing to happen is a paradox. Can you throw in a bit of mathematics, I'm interested.