
P(A)=\frac{n(A)}{n}
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
$\displaystyle \displaystyle P(A)=\frac{n(A)}{n}$.
Prove that this def. satisfies the three axioms of prob.
1 and 2 $\displaystyle 0\leq P(A)\leq P(S)=1\Rightarrow 0\leq P(A)\leq 1$
$\displaystyle \displaystyle P(A)\in [0,1]=\frac{n(A)}{n}\Rightarrow n\in [0,1]=n(A)\Rightarrow 0\leq n(A)\leq n=1$
3 $\displaystyle A_i\cap A_j=\emptyset, \ \ i\neq j$, then $\displaystyle \displaystyle P\left(\bigcup_{i=1}^{\infty}A_i\right)=\sum_{i=1} ^{\infty}P(A_i)$
Not sure what to do with 3.

For disjoint sets $\displaystyle A_1, ..., A_n$ it happens to be the case that, using your notation,
$\displaystyle
n\left(A_1 \cup \cdots \cup A_k\right) = \sum_{i = 1} ^ k n(A_i).
$
Note that because n < infinity, you can have at most n nonnull disjoint events so it suffices to show finite additivity as opposed to countably infinite.