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Say I have the following

$$\bigcap_{n \in \mathbb{N}} \left(-\frac{1}{n}, \frac{1}{n}\right)$$

What is the value of this intersection of an infinite amount of open sets?

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This intersection contains all real numbers $x$ such that $-1/n < x < 1/n$. Can you name such a number? Another one? – martini Oct 16 '12 at 20:01
I see you know the answer now. But you might be interested in this: try to find an infinite union of closed sets that is not closed. – mez Feb 5 '13 at 16:28
up vote 10 down vote accepted

HINT: There is exactly one real number in that intersection: what is it? Once you’ve identified it, you should go on to prove that if $x$ is any other real number, then there is some $n\in\Bbb Z^+$ such that $x\notin\left(-\frac1n,\frac1n\right)$.

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Cheers..I see it now..if I pick any number other than zero I can find an $n$ such that $x$ is not in that interval. – dukenukem Oct 16 '12 at 20:10
@dukenukem: Yep, that’s exactly right. – Brian M. Scott Oct 16 '12 at 20:13

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