Disjoint Set Proof Let A1, A2, ... be an arbitrary inﬁnite sequence of events, and let B1, B2, ... be another inﬁnite sequence of events deﬁned as follows: 
$B_1 = A_1, B_i = \frac {A_i}{\bigcup^{i-1}_{j=1} A_j}$ for i = 2, 3, ... 
Prove that B1,B2,... is a disjoint collection of events:
I have no idea how to tackle this? I understand what a disjoint collection is, but I don't know where to begin with this one?
Any help is appreciated

I'll sketch a proof. Use induction on $B_1,...,B_n$. $B_1$ by itself is obviously a family of disjoint sets so the base case is easy. Let $m=n+1$. If $B_1,...,B_n$ are disjoint, it shouldn't be too hard to show that $B_1,...,B_m$ are disjoint. Basically, $B_m$ is disjoint from the sets $A_1,...A_n$ which are supersets of $B_1,...B_n$. Then deduce for any  $B_i$ and $B_j$  with $i \neq j$, $B_i$ and $B_j$ are disjoint. This means that $B_1,...$ are pairwise disjoint.
Upon further thought, induction is just overkill. If you have natural numbers $i,j$ with $i \neq j$, then just pick the larger number. WLOG, assume $j>i$. Then $B_j$ is disjoint from $A_i$ by construction. $A_i$ is a super set of $B_i$. So the family is pairwise disjoint.
