Infinite Intersections of Infinite Sets

[MrBeezer]

Homework Statement

Decide if the following represents a true statement about the nature of sets. If it does not, present a specific example that shows where the statement does not hold:

If A_{1}\supseteqA_{2}\supseteqA_{3}\supseteqA_{4}\supseteq...A_{n} are all sets containing an infinite number of elements, then the intersection \bigcap^{\infty}_{n=1}A_{n} is infinite as well.

Homework Equations

The Attempt at a Solution

I decided to attempt this using proof by induction.. although I'm a little unsure of how to do this using sets, here it goes anyways:

Step 1:


Show that \bigcap^{k}_{n=1}A_{n} is infinite when k=1

\bigcap^{1}_{n=1}A_{n}=A_{1}

A_{1} is infinite.

Therefore,

\bigcap^{k}_{n=1}A_{n} is infinite when k=1

Next Step:

Let,
\bigcap^{k}_{n=1}A_{n} be infinite.

Inductive Step:

Show that \bigcap^{k+1}_{n=1}A_{n} is infinite

\bigcap^{k+1}_{n=1}A_{n}=\bigcap^{k}_{n=1}A_{n}\capA_{k+1}

\bigcap^{k}_{n=1}A_{n}\capA_{k+1}=A_{k+1}

A_{k+1} is infinite.


Therefore,


\bigcap^{k+1}_{n=1}A_{n} is infinite.

\bigcap^{\infty}_{n=1}A_{n} is infinite.





Are there any glaring errors here?

Any input would be greatly appreciated.

Thanks!

-Mike

 

[micromass]

\bigcap^{k+1}_{n=1}A_{n} is infinite.

\bigcap^{\infty}_{n=1}A_{n} is infinite.

That's a huge step here. I agree that induction shows that \bigcap_{k=1}^n{A_k} is infinite. But you cannot just conclude that the same holds for an infinite intersection!

If I were you, I'd start looking for a counter-example.

 

[MrBeezer]

Ok, that's what I was afraid of. Thank you for pointing out the gap there. I think I know a counter example, but my flawed inductive proof influenced me not to try it.

Let A_{1}= {1,2,3,4...}
A_{2}={2,3,4,5...}
A_{3}={3,4,5,6...}

Assume there is an element, k\in\bigcap^{\infty}_{n=1}

k\notin\bigcap^{k+1}_{n=1} A_{n}

therefore, \bigcap^{\infty}_{n=1} A_{n}= (empty set).

Ok, so this specific example shows an infinite intersection of infinite sets that equals the empty set( I think), thus disproving the statement above.

is there a way to show that this is true for all infinite intersections of infinite nested subsets? Or can you get different results depending on the type of infinite sets..

 

[Dick]

Ok, that's what I was afraid of. Thank you for pointing out the gap there. I think I know a counter example, but my flawed inductive proof influenced me not to try it.

Let A_{1}= {1,2,3,4...}
A_{2}={2,3,4,5...}
A_{3}={3,4,5,6...}

Assume there is an element, k\in\bigcap^{\infty}_{n=1}

k\notin\bigcap^{k+1}_{n=1} A_{n}

therefore, \bigcap^{\infty}_{n=1} A_{n}= (empty set).

Ok, so this specific example shows an infinite intersection of infinite sets that equals the empty set( I think), thus disproving the statement above.

is there a way to show that this is true for all infinite intersections of infinite nested subsets? Or can you get different results depending on the type of infinite sets..

Good job. That's an infinite intersection of infinite sets that's empty. And sure, the result depends on the sets. Can you give an example where the infinite intersection of infinite sets is infinite?

 

[MrBeezer]

Well, don't have to use proper subsets so technically we could have nested subsets where A_{1}=A_{2}=A_{3}=A_{4}... right? I know the infinite intersection of those sets would be infinite, but that's no fun.

What if we had a infinite intersection of sets where A_{n}=N-\sum^{n}_{1}2n

So,

A_{1}={1,3,4,5,6,7,8...}
A_{2}={1,3,5,6,7,8...}
A_{3}={1,3,5,7,8...}

In this case,

\bigcap^{\infty}_{n=1}={1,3,5,7,9,11...}

So the infinite intersection equals an infinite amount of odd natural numbers.

 

[SammyS]

Well, don't have to use proper subsets so technically we could have nested subsets where A_{1}=A_{2}=A_{3}=A_{4}... right? I know the infinite intersection of those sets would be infinite, but that's no fun.

What if we had a infinite intersection of sets where A_{n}=N-\sum^{n}_{1}2n

So,

A_{1}={1,3,4,5,6,7,8...}
A_{2}={1,3,5,6,7,8...}
A_{3}={1,3,5,7,8...}

In this case,

\bigcap^{\infty}_{n=1}A_n = {1,3,5,7,9,11...}

So the infinite intersection equals an infinite amount of odd natural numbers.

No doubt you mean: \displaystyle A_{n}=\mathbb{N}-\bigcup_{k=1}^n\{2k\}

 

[MrBeezer]

Yes I do, thank you!

 

[Gregory S.]

One proof of a version of Cauchy's theorem actually uses the fact that this is not the case. You take the intersection of some specific nested triangles in the complex plane, which shrink to exactly one point. It seems like you had countably infinite sets in mind, but this proof is too pretty to not mention.