When Cantor's Intersection theorem won't work with closed sets

Give an example to show that Cantor's Intersection Theorem would not be true if compact sets were replaced by closed sets.

Compact set is closed and bounded, so what I'm going to find is something that is closed but not bounded.

By the Cantor theorem, which says that a decreasing sequence of non-empty compact subset will have a non-empty intersection.

What I was trying to approach is how two sets just "touched" and as if they have not bounded, two sets "touched" nothing. But I cant give an example for that

Solution 1:

Consider the intersection of all sets of the form $[n,\infty)$, where $n$ ranges over the positive integers.