Compact Nested Sequences and Their Intersection

[JamesF]

Hi everyone. I feel like I'm really close to the answer on this one, but just out of reach :) I hope someone can give me some pointers

Homework Statement

Let A1 \supseteq A2 \supseteq A3 \supseteq \ldots be a sequence of compact, nonempty subsets of a metric space (X, d). Show that \bigcap A_n \neq \emptyset. (Hint: Let U_n = X-A_n)

The Attempt at a Solution

I tried showing by contradiction.

Suppose \bigcap A_n = \emptyset
Choose an open subcover U_n = X-A_n (that's supposed to be set minus but I don't know how to do \ in tex). Then \bigcup U_n = (X-A_1) \cup (X - A_2) \ldots = X - (\bigcap A_n) = X

but where's the contradiction? So X is not compact, but that goes without saying and we can't infer much from that. What am I overlooking here? Or is this the wrong approach entirely?

Thank you for your assistance.

 

[Dick]

X may not be compact but A1 is. The U_n are a cover of A1 if you assume the intersection of all the Ai is empty. Hence there is a finite subcover. That seems headed for a contradiction.

 

[JamesF]

Thanks for your help, Dick. I was able to get the solution.


I have one more question on my current HW.
Is a set A_n = [n, \infty) open or closed in \mathbb{R}? I would think so, but it's unbounded.

 

[Dick]

Thanks for your help, Dick. I was able to get the solution.


I have one more question on my current HW.
Is a set A_n = [n, \infty) open or closed in \mathbb{R}? I would think so, but it's unbounded.

What do you think it would be?