Infinite intersection of indexed sets

[math771]

Consider the set A_n=\{0.9, 0.99, 0.999,...\} , where the greatest element of A_n has n 9s in its decimal expansion. Then 0.999\ldots=1\in\bigcap_{n=1}^\infty{A_n}. Is this possible even though \not\exists{n}(1\in{A_n})?

Edit: I see that 0.999\ldots=1\not\in\bigcap_{n=1}^\infty{A_n}. Sorry :(.

[HallsofIvy]

It is quite common that the limit of a sequence has some property that no member of the sequence has. Nothing at all strange about that.