I am trying to solve the following problem in Kolmogorov's real analysis textbook.
Give an example of a complete metric space $R$ and a nested sequence $\{A_n\}$ of closed subsets of $R$ such that $\bigcap\limits_{n=1}^{\infty} A_n = \emptyset$. Reconcile this example with [Cantor's intersection theorem.]
Cantor's intersection theorem states that for a complete metric space $R$ and a sequence $\{A_n\}$ of closed and nested subsets whose diameter $0$ as $n \to \infty$, the infinite intersection of the $A_n$ is nonempty.
Based on the statement of this theorem, it seems that the only assumption I do not have is the fact that the diameter of the $A_n$ approaches $0$. It would seem, therefore, that this is how we get an example that we can reconcile with Cantor's intersection theorem, but I am still struggling to think of a good example. If $R$ didn't have to be complete, I think it would certainly be possible to find such a set. But I recall from the proof of Cantor's intersection theorem that the diameter approaching $0$ is needed to establish that the sequence of elements generated from the $A_n$ is Cauchy. I'm not sure if this lends itself to a proof via constructing a sequence of $A_n$ whose elements do not become infinitely close to each other.
Any help on this result would be greatly appreciated.
Take $A_n=[n,+\infty)$ with the usual metric on the real line.