Cantor's Theorem states: "Suppose that X is a non-empty complete metric space, and $C_n$ is a sequence of closed nested subsets of X whose diameters tend to zero:
$\lim_{n \rightarrow \infty} diam(C_n) = 0$
Where $diam(C_n) = sup \{d(x,y): x,y \in C_n\}$
Then the intersection of the $C_n$'s contains exactly one point:
$\cap^{\infty}_{n=1} C_n = \{x\}$."
For an assignment I have to show that the hypothesis that the diameter tends to zero is needed for the theorem by showing if $\lim_{n \rightarrow \infty} diam(C_n) \neq 0$ the intersection becomes empty.
This seems incorrect to me because intuitively if diameter of set is always non-zero then a set must contain at least two distinct points, and the intersection of nested sets which always contain at least two points is should not be empty. Right?
My suspicion is that my intuition about this is incorrect, because it's more likely that I am wrong than that the question is.
Can someone help me understand this better? Thank you for your help.
Best Answer
The claim should be that the intersection can be empty if this condition is dropped, not that it always will be.
Consider $X = \mathbb{N}$ with the discrete metric, and $C_n = \{n, n+1, n+2, \dots\}$.
A similar example would be $X=\mathbb{R}$ and $C_n = [n,+\infty)$.
Just because each of the sets $C_n$ contains at least two points (infinitely many, in these examples), does not mean that any one of those points has be common to all the sets, which is what would be necessary for the intersection to be nonempty.