Question regarding the finite intersection property Suppose I have a sequence $\{C _j \} $ of closed and bounded intervals such that $ \cap _{j=1 } ^{\infty } C _j= \emptyset$. Does it follow that for some $n $, we must have $\cap _{j } ^n E _j = \emptyset$
I believe this is the only possibility since otherwise the assumption would violate the finite intersection property.

Yes, this is correct in a space where closed and bounded intervals are compact. If every finite intersection were nonempty, this would also be true when intersect all the other intervals with a single interval. In that case compactness gives us the result.
