I just need a quick clarification:

Given a sequence of sets $\{a_n\}_{n \in \mathbb{N}}$ in some field $\mathbb{K}$, is saying that it satisfies the finite intersection property equivalent to saying $(\forall n\in \mathbb{N})(\exists x\in \mathbb{K})(x \in \cap_{i=1}^n a_n)$

If the previous statement is true, then it seems almost reasonable to say that, because $\{\cap_{i=1}^n a_n\}_{n\in\mathbb{N}}$ is a nested sequence of nonempty sets (because of the f.i.p), $\cap_{i=1}^{\infty} a_n \neq \emptyset$, but I know that is not necessarily true

Thanks

`$A_n=\{m\in\mathbb{N}:m>n\}$`

. This is really a bit too basic for MO. – Robin Chapman Jun 23 '10 at 15:51