Just take something from the last subset

Does there exist an infinite family of non-empty sets $A_i$ with $A_1=[0,1]$ (i.e all the number between 0 and 1 inclusive) which satisfies

$A_n \subseteq A_{n-1}$

such that there doesn't exist $X$ which is in all of the subsets , i.e $\forall x \in \mathbb{R}, \exists \space n \space s.t \space x \space \notin A_n$ ?