TITLE: What is $\bigcap_{n \in \mathbb{N}} \left(0, {1\over n}\right)$?
QUESTION [5 upvotes]: What is$$\bigcap_{n \in \mathbb{N}} \left(0, {1\over n}\right)?$$I suspect it is the empty set, and we would see this by using the Archimedean property of $\mathbb{R}$ or something like that, but I have no idea on how to prove it. Can anybody help me? Thanks in advance!

REPLY [4 votes]: You got it right: it is indeed an empty set. To prove this, suppose it isn't empty, that is $\exists x\in\mathbb{R},\,x\in\bigcap\limits_{n\in\mathbb{N}}\left( 0,\frac{1}{n}\right)$. Since there's an intersection, this means that $\forall n\in\mathbb{N},\,x\in\left( 0,\frac{1}{n}\right)$, that is $x>0$ and $\forall n\in\mathbb{N},n<\frac{1}{x}$. Guess you found the contradiction :D