Intersection of sets with infinite number of elements

[autre]

I have to decide whether the following is true or false:

If A1$\supseteq$A2$\supseteq$A3$\supseteq$...are all sets containing an infinite number of elements, then the intersection of those sets is infinite as well.

I think I found a counterexample but I'm not sure the correct notation. I to have sets {n, n+1, n+2,...} from n to infinity (so {1, 2, 3,...}$\supseteq${2,3,4,...}) and the intersection of those sets is obviously null. How do I write this out? Thanks!

 

[micromass]

Just write it as

$$A_n=\{n,n+1,n+2,...\}$$

then

$$\bigcap_{n\in \mathbb{N}}{A_n}=\emptyset$$

 

[disregardthat]

The intersection of a set of sets is the set of elements contained in every of those sets. What number is contained in every such set? (Hint: assume n is in the intersection, and find a set which does not contain n)

 

[autre]

Thanks micromass, that's the notation I was looking for.

 

[blue_raver22]

Your counterexample is correct. In notation, you can write it as:

A1 = {n, n+1, n+2, ...}
A2 = {n+1, n+2, n+3, ...}
A3 = {n+2, n+3, n+4, ...}

The intersection of these sets would be:
A1 ∩ A2 ∩ A3 = {n+2, n+3, n+4, ...} ∩ {n+1, n+2, n+3, ...} ∩ {n, n+1, n+2, ...}

= {n+2, n+3, n+4, ...} ∩ {n+2, n+3, n+4, ...}

= {n+2, n+3, n+4, ...}

Which is just the set containing elements from n+2 to infinity, and thus has an infinite number of elements. Therefore, the statement "the intersection of sets with an infinite number of elements is also infinite" is false, as your counterexample proves.