Cantor intersection theorem I have Cantor intersection theorem:
Let $X$ be a complete metric space, and let $\{F_n\}$ be a decreasing sequence of non-empty closed subsets of X such that $d(F_n)$ converges to $0$.
Then $F=\bigcap_{n=1}^\infty F_n$ contains exactly one point.
What affects whether the condition $d(F_n)$ converges to zero is removed?
Give an example to show that the set $F$ in the intersection of Cantor theorem may be empty if this happens?

The easiest (and I think necessary in some sense) type of example where d(Fn) does not converge to 0 and the intersection is void is when d(Fn) = infinity.
For instance R and the intervals [n,infinity) are all closed, yet their intersection is empty. Besides if you didn't have d(Fn) to 0 you wouldn't get a unique point (like say d(Fn)--->c < infinity)
