I'm trying to prove Boole's inequality
$$P\left(\ \bigcup_{i=1}^\infty A_i\right) \leq \sum_{i=1}^\infty P(A_i)$$
by constructing $\{B_1, B_2,..., B_n\}$ such that $F_0 = \emptyset$, $F_n =\bigcup_{i=1}^n A_i$ and $B_i = A_{i} - F_{i-1}$. I have showed by induction that for $n \geq 1$, we have
$$ \bigcup_{i=1}^n A_i = \bigcup_{i=1}^n B_i.$$
Since for all $i\neq j$, $B_i \cap B_j = \emptyset$, therefore $\{B_1, B_2,..., B_n\}$ is a partition of $F_n$.
I am stuck at the step where I must show that this is true for $\infty$:
$$ \bigcup_{i=1}^\infty A_i = \bigcup_{i=1}^\infty B_i.$$