Increasing sequence of events and the probability of their limit 
If $A_1\subset A_2\subset A_3\subset\cdots$ is an increasing sequence of events with limit $A=\bigcup_{i=1}^\infty A_i$. Prove that $$\lim_{n\rightarrow\infty} P(A_n)=P(A)$$

My attempt so far:
Since $A_1\subset A_2\subset A_3\subset\cdots$ is increasing and $$\bigcup_{i=1}^\infty A_i=\lim_{n\rightarrow\infty} A_n \wedge A=\bigcup_{i=1}^\infty A_i$$ then $\lim_{n\rightarrow\infty}A_n=A$. I have that $$P\left(\lim_{n\rightarrow\infty}A_n\right)=P(A)\Rightarrow \lim_{n\rightarrow\infty}P(A_n)=P(A)$$

Hint: One usually requires from a probability to be $\sigma$-additive, i.e. when $(B_i)_{i\geq 1}$ is a countable disjointed family, then $P(\bigcup_i B_i)=\sum_i P(B_i)$. So try to rewrite your increasing union as a countable union of disjoint sets.
