Simple probability proof about limits

[hermanni]

Hi all,
I'm trying to solve the 5th question , it's from Allan Gut's probability : a gradute course .I attached the relevant pages.
For the part (a) I think we don't need to construct the sequence itself, but prove it exists somehow.
For the part (b) we need to find an example.For the parts (b) and (c) , I really don't have an idea. Can anyone help please? Thanx.

 

[jgm340]

I'll give you a hint for part (a):

Let {E_n} be a sequence of sets. For each n, let C_n be the compliment of E_n.

Suppose that P(C₁) = a, P(C₂) = a/2, P(C₃) = a/4, etc. In other words, P(C_n) = a/2^(n-1).

What can we say then about P($$\bigcup$$ C_n)?

Remember that 1/2ⁿ is a geometric series. Tell me, what the sum from 0 to infinity of 1/2ⁿ? What about a/2ⁿ?Also, if we know P($$\bigcup$$ C_n), then don't we know P($$\bigcap$$ E_n)?

Now, try to assemble these facts together!

 

[hermanni]

Thank you very much for this construction What about part c? How can we show such a thing?

Regards, h.