Intersection of infinitely many sets being none of them In my analysis class it came up that
$\bigcap_{q\in \mathbb{N}}[0,1/q] = \{0\}$
He was easily able to prove that this was so with a proof by contradiction since for any number greater than zero there is a $1/q$ less than that number.
However, we can also say that for the intersection of two sets when one is a subset of the other is going to be the set that is the subset of the others.
We can also say that for every two intervals of the form $[0,1/q]$, one of them is a subset of the other.
These two things together seem to imply that $\bigcap_{q\in \mathbb{N}}[0,1/q]$ must be contained in the set $\{[0,1/q],q \in \mathbb{N}\}$ which $\{0\}$ certainly is not.
Why is the conclusion false and not creating a contradiction.

This is an example of how infinite collections behave very differently from finite ones.
You can show that if a finite collection of sets has the property that each is a subset or a superset of any other, then the intersection of the sets in that collection is one of the sets in that collection. But that says nothing about the infinite case.
Let me take a simpler example, with the same underlying logic. Any finite set of reals has a least element, but the set of positive reals has no least element.
The moral: when you want to prove something about infinite sets, it's not enough to just prove it for finite sets. Infinite sets have properties which finite sets don't.
