# Infinite intersection of indexed sets

1. Jul 16, 2011

### 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 :(.

Last edited: Jul 16, 2011
2. Jul 17, 2011

### 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.