Hi,

I'm a real dunce when it comes to indexed sets/Collections etc. I can't seem to understand this question at all, so I would really appreciate some help with the following question.

Let Ak is a set for each positive integer k. Define another collection of sets Bk:

B1 := A1, B2 := A2 \ A1, · · · , Bk = Ak \ Ut<kAt

, · · · .

Prove that

(a) The sets Bk are pairwise disjoint;

(b) Ut≤kAt = Ut≤kBt for each positive integer k;

(c) Ut≥1At = Ut≥1Bt

.