For disjoint sets it happens to be the case that, using your notation,
Note that because n < infinity, you can have at most n non-null disjoint events so it suffices to show finite additivity as opposed to countably infinite.
Suppose that an experiment is performed n times. For any event A of the experiment, let n(A) denote the number of times that event A occurs. The relatively frequency definition of probability would propose that
.
Prove that this def. satisfies the three axioms of prob.
1 and 2
3 , then
Not sure what to do with 3.