Let $\displaystyle \{A_1, A_2, A_3, ...\}$ be a sequence of events of a sample space $\displaystyle S$. Find a sequence $\displaystyle \{B_1, B_2, B_3, ...\}$ of mutually exclusive events such that for all $\displaystyle n \geq 1$, $\displaystyle \bigcup_{i=1}^m A_i = \bigcup_{i=1}^n B_i$.