Understanding of countable set short theorem. I am struggling to understand what the theorem is stating. 

Question 1
Let n equals 2, would that mean that $B_n$ would be $B_2$= $\left\{\left\{a_1\right\},\left\{a_1,a_2\right\}\right\}$?
Question 2
Given n=2 and 3 how would small b, be described? 
In the case of $B_{n-1}$,would the sets (b,a), be
(0,$a_1$),($b_1$,$a_2$).
I have a feeling I am totally lost and I am hoping someone could provide a descret example so that I understand what it is stating?

My answer for Question 1 
Let A = {a,b,c} then B_2 = {(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c)}
That's way the proof says B_1 = A 
