Nested interval theorem for R^n

In summary, the nested interval theorem for R^n states that a decreasing sequence of bounded closed sets in R^n, where each set is non-empty, will have a non-empty intersection. This is proven by considering a sequence of points (p_i : i in N) formed from the non-empty sets, which is bounded and has a convergent subsequence by the Bolzano-Weierstrass theorem. Due to the nested nature of the sets, the limit point of this subsequence is in the intersection of all sets, proving the theorem.
  • #1
eckiller
44
0
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?
 
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  • #2
The key is these sets are nested. So [tex]p_i \in K_i \subset K_{i-1} \subset K_{i-2} \subset \ldots \subset K_1[/tex]

So for [tex]j\ge i[/tex], [tex]p_j \in K_i[/tex]. So the tail of the convergent subsequence is in [tex]K_i[/tex]. So the limit in is [tex]K_i[/tex] 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 [tex]K_i[/tex]'s so it is in the intersection.

Hope that helps,
Steven
 
  • #3
Excellent, many thanks.
 

Related to Nested interval theorem for R^n

What is the Nested Interval Theorem for R^n?

The Nested Interval Theorem for R^n states that if a sequence of closed, bounded intervals in R^n satisfies the property that the length of each interval approaches zero as the sequence progresses, then there exists a single point in the intersection of all of the intervals.

What is the significance of the Nested Interval Theorem for R^n?

The Nested Interval Theorem for R^n is significant because it provides a way to prove the existence of points in the intersection of multiple intervals, even when the intervals are getting smaller and smaller. This is useful in many areas of mathematics, including real analysis and topology.

How is the Nested Interval Theorem for R^n different from the Nested Interval Theorem for R?

The Nested Interval Theorem for R^n is a generalization of the Nested Interval Theorem for R, which only applies to intervals in one dimension. The theorem for R^n applies to intervals in n-dimensional space, making it more versatile and useful in higher dimensions.

Can the Nested Interval Theorem for R^n be extended to infinite sequences of intervals?

Yes, the Nested Interval Theorem for R^n can be extended to infinite sequences of intervals as long as the intervals satisfy the same property that their lengths approach zero as the sequence progresses. This is known as the Nested Interval Property.

What are some applications of the Nested Interval Theorem for R^n?

The Nested Interval Theorem for R^n has applications in various fields of mathematics, including real analysis, topology, and measure theory. It is also used in the proof of other theorems, such as the Baire Category Theorem and the Bolzano-Weierstrass Theorem.

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