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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
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1 Answer 1

up vote 9 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
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@dukenukem: Yep, that’s exactly right. –  Brian M. Scott Oct 16 '12 at 20:13
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