Cantor's intersection theorem (Apostol)

In summary, the conversation discusses the author's proof of a theorem in "Mathematical Analysis" by Apostol. The author assumes that the nested sets in the proof contain infinitely many points, otherwise the proof would be trivial. The question is raised as to why this is the case and how to prove it. The expert summarizer explains that if a finite set is nested within an infinite set, the number of elements in the sets will eventually reach a point where they will not decrease any further. At this point, all sets in the sequence will be equal, making the intersection of the sets non-empty. The conversation concludes with the OP expressing gratitude for the clarification.
  • #1
mr.tea
102
12
Hi,

I am reading "mathematical analysis" by Apostol right now for a course in analysis. Since I am trying to understand the author's proof of the above theorem(3.25 in the book), but I have something that I can't understand.
He assumes that each of the nested sets contains infinitely many points, "...otherwise, the proof is trivial". I can't see why it's trivial and how to prove it. I would be grateful to understand why an infinite intersection of finite sets is non-empty.

Thank you.
 
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  • #2
mr.tea said:
Hi,

I am reading "mathematical analysis" by Apostol right now for a course in analysis. Since I am trying to understand the author's proof of the above theorem(3.25 in the book), but I have something that I can't understand.
He assumes that each of the nested sets contains infinitely many points, "...otherwise, the proof is trivial". I can't see why it's trivial and how to prove it. I would be grateful to understand why an infinite intersection of finite sets is non-empty.

Thank you.
You have the non-empty nested sets ##C_0 ⊇ C_1 ⊇ ... ⊇ C_k ⊇ ...##.
Assume ##C_k## is finite for some k. Then all the ##C_j## with ##j>k## will also be finite, and equal to or smaller than ##C_k##. As the number of elements in the sets of the sequence is decreasing, at a certain point in the sequence the number of elements will decrease no more (as it can't become 0).
From that point, all ##C_j## will be equal.
Hence the intersection will trivially not be empty
 
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  • #3
Samy_A said:
You have the non-empty nested sets ##C_0 ⊇ C_1 ⊇ ... ⊇ C_k ⊇ ...##.
Assume ##C_k## is finite for some k. Then all the ##C_j## with ##j>k## will also be finite, and equal to or smaller than ##C_k##. As the number of elements in the sequence is decreasing, at a certain point in the sequence the number of elements will decrease no more (as it can't become 0).
From that point, all ##C_j## will be equal.
Hence the intersection will trivially not be empty

Wow, thank you. What I missed was your note in the parenthesis. I feel embarrassed not noticing it.

Thank you.
 

FAQ: Cantor's intersection theorem (Apostol)

What is Cantor's intersection theorem (Apostol)?

Cantor's intersection theorem, also known as the Nested Interval Theorem, is a fundamental theorem in real analysis that states that if a sequence of nested closed intervals in the real numbers has a non-empty intersection, then the intersection contains exactly one point.

What is the significance of Cantor's intersection theorem (Apostol)?

Cantor's intersection theorem is significant because it provides a rigorous mathematical proof for the existence of a real number within a set of nested intervals. It is also used in many other theorems and proofs in real analysis.

What are nested intervals?

Nested intervals are a sequence of closed intervals, where each interval in the sequence contains the previous interval and the length of the intervals approaches zero as the sequence progresses.

What is the proof of Cantor's intersection theorem (Apostol)?

The proof of Cantor's intersection theorem involves using the completeness property of the real numbers, which states that every non-empty set of real numbers that is bounded above has a least upper bound. By constructing a sequence of nested intervals, it can be shown that the intersection of these intervals contains the least upper bound, thus proving the theorem.

Can Cantor's intersection theorem (Apostol) be extended to higher dimensions?

Yes, Cantor's intersection theorem can be extended to higher dimensions. In higher dimensions, the nested intervals become nested closed sets and the theorem holds as long as the sets are bounded and have non-empty intersections. This is known as the generalized nested interval theorem.

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