# Cantor's Intersection Theorem with closed sets [duplicate]

Cantor's Intersection Theorem states that "if $\{C_k\}$ is a sequence of non-empty, closed and bounded sets satisfying $C_1 \supset C_2 \supset C_3 \dots$, then $\bigcap_{n \ge 1} C_n$ is nonempty.

If the term "compact sets" is replaced by "closed sets", the statement is not true. It makes sense to me, but couldn't find such a counterexample for it.

## marked as duplicate by user228113, Claude Leibovici, user91500, Paul Plummer, N. F. TaussigJan 31 '16 at 9:38

• You can create an ellipsis in MathJax by using \dots. – Michael Albanese Jan 31 '16 at 5:18
Consider the sequence $C_n = [n, \infty)$.
• There is no real number which belongs to $C_n$ for every $n$, so the intersection is empty. – Michael Albanese Jan 31 '16 at 5:27