Problem in Chapter 2 (Walter Rudin). "Principles of Mathematical Analysis" by Walter Rudin has the following question:

Show that the following statement is false in $\Bbb{R}$:
If $\{K_\alpha\}$ is a collection of closed subsets of a metric space $X$ such that the intersection of every finite subcollection of $\{K_\alpha\}$ is nonempty, then $\bigcap K_\alpha$ is nonempty.

If every finite intersection is nonempty, this implies we can construct nested closed sets. Considering $\Bbb{R}$ is complete, shouldn't this intersection be nonempty? Why is this statement false?

The example given in the other answer shows why it is false. Note that the diameters of these sets are all infinite. If indeed we have countably many such closed sets and their diameters tend to $0$ as well, then completeness will give a point in the intersection (as picking points in the intersections will give a Cauchy sequence). Maybe this is what confused you?
