How to Prove the Nested Interval Theorem for R^n?

[eckiller]

Hi,

I am trying to prove the nested interval theorem for R^n. It is stated as follows:

Let (K_i : i in N) be a decreasing sequence of bounded closed sets in R^n and each K_i is non empty. Then the intersection of all K_i for i in N is not empty.

This is what I have so far:

Since K_i is nonempty, there exists a p_i in K_i for i in N. Pick these points to form the sequence (p_i : i in N). Because (p_i : i in N) is bounded, it has at least one convergent subsequence by the Bolzano-Weierstrass theorem.

I want to claim that the limit point of this subsequence is in the intersection of all K_i. But I am having trouble formalizing this part of the argument. Any help?

 

[snoble]

The key is these sets are nested. So $$p_i \in K_i \subset K_{i-1} \subset K_{i-2} \subset \ldots \subset K_1$$

So for $$j\ge i$$, $$p_j \in K_i$$. So the tail of the convergent subsequence is in $$K_i$$. So the limit in is $$K_i$$ since it is closed. Since this is true for an arbitrary i it is true for all i. So the limit is in all of the $$K_i$$'s so it is in the intersection.

Hope that helps,
Steven

 

[eckiller]

Excellent, many thanks.