For any countable events (A_1, A_2, dots), define
(B_1 := A_1)
(B_2 := frac{A_2}{A_1})
(vdots)
(B_n := frac{A_n}{(A_1 cup cdots cup A_{n-1})})
(vdots)
Then, prove that
(a) ({B_n}_{n=1}^{infty}) are pairwise disjoint,
(b) (B_n subseteq A_n),
(c) (cup_{k=1}^{n} B_k = cup_{k=1}^{n} A_k) for all (n), and
(d) (cup_{n=1}^{infty} B_n = cup_{n=1}^{infty} A_n).
4. For any countable events (A, A_2, dots), define
(B = A, B = A, )
(B_n = A_n cap (A_1 cup cdots cup A_{n-1}))
(dots)
Then, prove that
(a) (B) are pairwise disjoint, (b) (B_n subseteq A_n), (c) (cup_{n=1} B = cup A) for all (n) and (d) (cup_{n=1} B = cup A)
