Q: Decide which statements are true or false. If false, give an example of where it falls short.
a) If are all sets containing an infinite number of elements, then the intersection is infinite as well.
b) If are all sets containing an finite number of elements, then the intersection is finite and nonempty.
a) True. I figure this is true, because every infinite interval on the real line can be broken down into a smaller interval; so, all the sets will tend towards some kind of singularity, but this will also be infinite.
b) False: suppose some element is in the intersection, we are saying this element must be in every single . This is clearly not true for . This is assuming .
Are my answers correct?