Nested sequence of compact sets in Rn has a non-empty intersection?

[seeker101]

There's a theorem that says any nested sequence of compact sets in Rn always has a non-empty intersection. So there is something wrong with this counterexample. I'm not able to see what's wrong:

Consider the interval Un = [2-1/n, 1+1/n] for n=1, 2 and 3.
Isn't the intersection of U1, U2 and U3 the null set? (since U3 is the null set?)

 

[HallsofIvy]

There's a theorem that says any nested sequence of compact sets in Rn always has a non-empty intersection. So there is something wrong with this counterexample. I'm not able to see what's wrong:

Consider the interval Un = [2-1/n, 1+1/n] for n=1, 2 and 3.
Isn't the intersection of U1, U2 and U3 the null set? (since U3 is the null set?)

There is a theorem that say any nested sequence of non-empty compact sets in Rⁿ is has non-empty intersection.