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?

Because $\infty$ is not part of the real numbers. So set
$$
K_\alpha = [\alpha, \infty)
$$
