Cantor's intersection theorem (Apostol)

[mr.tea]

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.

 

[Samy_A]

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

 

[mr.tea]

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.