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 \(\forall x \in \mathbb{R}, \exists \space n \space s.t \space x \space \notin A_n\)?

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