1. ## Nested set question

Q: Decide which statements are true or false. If false, give an example of where it falls short.

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

b) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an finite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is finite and nonempty.

A:

a) True. I figure this is true, because every infinite interval on the real line can be broken down into a smaller interval; so, all the sets will tend towards some kind of singularity, but this will also be infinite.

b) False: suppose some element $m$ is in the intersection, we are saying this element must be in every single $A_{n}$. This is clearly not true for $m+1\in\\\mathbb{N}$. This is assuming $A_{n}=\{n,n+1,n+2,...\}$.

Thank you!

2. Hello Danneedshelp
Originally Posted by Danneedshelp
Q: Decide which statements are true or false. If false, give an example of where it falls short.

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

b) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an finite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is finite and nonempty.

A:

a) True. I figure this is true, because every infinite interval on the real line can be broken down into a smaller interval; so, all the sets will tend towards some kind of singularity, but this will also be infinite.

b) False: suppose some element $m$ is in the intersection, we are saying this element must be in every single $A_{n}$. This is clearly not true for $m+1\in\\\mathbb{N}$. This is assuming $A_{n}=\{n,n+1,n+2,...\}$.

Thank you!
I'm afraid I don't understand your answer to (b). Your definition $A_{n}=\{n,n+1,n+2,...\}$ appears to define sets containing infinitely many elements.

The question as you have stated it doesn't insist that all the sets are non-empty. So if $\exists j, A_i = \{\}, i \ge j$, then the intersection $\bigcap_{n=1}^{\infty}=\{\}$

But if all the sets are finite and non-empty, then isn't it the case that, after a certain point all the sets must be equal; i.e. $\exists j, A_i=A_j, i>j$? In which case, this 'limiting' set will be the intersection.

a) If $A_{1}\supseteq\\A_{2}\supseteq\\A_{3}\supseteq\\A_ {4}...$ are all sets containing an infinite number of elements, then the intersection $\bigcap_{n=1}^{\infty}=A_{n}$ is infinite as well.
$A_n = \left( {0,\frac{1}{n}} \right)\; \Rightarrow \;\bigcap\limits_n {A_n } = \emptyset$