(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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[itex]_{1}[/itex][itex]\supseteq[/itex]A[itex]_{2}[/itex][itex]\supseteq[/itex]A[itex]_{3}[/itex][itex]\supseteq[/itex]A[itex]_{4}[/itex][itex]\supseteq[/itex]...A[itex]_{n}[/itex] are all sets containing an infinite number of elements, then the intersection [itex]\bigcap[/itex][itex]^{\infty}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite as well.

2. Relevant equations

3. 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 [itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite when k=1

[itex]\bigcap[/itex][itex]^{1}_{n=1}[/itex]A[itex]_{n}[/itex]=A[itex]_{1}[/itex]

A[itex]_{1}[/itex] is infinite.

Therefore,

[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite when k=1

Next Step:

Let,

[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex] be infinite.

Inductive Step:

Show that [itex]\bigcap[/itex][itex]^{k+1}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite

[itex]\bigcap[/itex][itex]^{k+1}_{n=1}[/itex]A[itex]_{n}[/itex]=[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex][itex]\cap[/itex]A[itex]_{k+1}[/itex]

[itex]\bigcap[/itex][itex]^{k}_{n=1}[/itex]A[itex]_{n}[/itex][itex]\cap[/itex]A[itex]_{k+1}[/itex]=A[itex]_{k+1}[/itex]

A[itex]_{k+1}[/itex] is infinite.

Therefore,

[itex]\bigcap[/itex][itex]^{k+1}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite.

[itex]\bigcap[/itex][itex]^{\infty}_{n=1}[/itex]A[itex]_{n}[/itex] is infinite.

Are there any glaring errors here?

Any input would be greatly appreciated.

Thanks!

-Mike

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Prove the intersection of nested subsets containing infinite elements is infinite

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