# Can an infinite sequence of nested closed intervals have common points (plural)

My question comes from an exercise in Stillwell's "Real Numbers," Ex. 3.5.8 0n page 73 regarding Cantor's first uncountablity proof. There is an excelent discussion of the proof here: Cantor's First Proof that $\Bbb{R}$ is uncountable

In the text, there is an introductory description where a sequence of nested closed intervals is obtained from consideration of a sequence of countable many real numbers.

The question is the asked to conclude that any of the common points of the infinite sequence of nested closed intervals are not equal to any of the points of the given sequence of real numbers.

I know an infinite sequence of nested closed intervals has one common point. I would appreciate clarification as to the possibility of more than one common point.

Thanks

EDIT I just realized I neglected to say closed intervals. (If that would alter Ross Milikan's answer)

If the lengths of the closed intervals converge to zero, there is just one common point. If not, the intersection is an interval itself and in the reals you have continuum many common points. For example our intervals could be indexed by $n$ and be $[-\frac 1n,1+\frac 1n]$ where the intersection is $[0,1]$
• Note that there need not even be a common point with open intervals: consider, for example, $\{(0, 1/n)\}_{n \in \mathbb N_+}$ Commented Jul 3, 2017 at 14:52