I can't really think of a real life example...but then I try not to worry about real life when I'm in math. But I'll try to be helpful to doing this problem.

Have you done proofs before? "if and only if" proofs? If so, then you know that one way to do them is to suppose the first half is true, then show (ie prove) that the second half follows. Then conversely you suppose the second half is true and then show that the first half follows.

Here's how I'd do part b)

Suppose that omega is in the intersection of B1,B2,... [we wish to show that omega is in an infinite number of the events A1,A2,...]

Just suppose that omega is in only a finite number of the events Ai. Then there is some integer N such that omega is NOT in A_N+1, A_N+2,... This would mean that omega is NOT in B_N+1, nor in B_N+2, nor in B_N+3, etc.

[now you have a contradiction, because of what you first supposed]

But this is impossible since omega is in the intersection of B1,B2,... Therefore omega must be in an infinite number of the events Ai.

On the other hand, now suppose that omega is in an infinite number of the events Ai. Then omega must be in B_N for any integer N. [look at the way B_n is defined]. Therefore omega is in the intersection of B1,B2,...