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I hope everyone who has underwent a fundamental course in Analysis must be knowing about Cantor's completeness principle. It says that in a nest of closed intervals ,the intersection of all the intervals is a single point. I hope I can get an explanation as to why in case of only closed intervals this principle holds good, why not in case of open intervals?

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Consider the nested intervals, $[0, 1 + \frac{1}{n}]$ for $n \in \omega$. The intersection is not a single point. – William Sep 19 '11 at 17:44
@SrivatsanNarayanan - I am sorry for landing up at such a mistake. Now when you have stated it correctly could you kindly answer as to why the word "closed" has importance attached to it. Why cannot we have the result holding good in case of open intervals? – Primeczar Sep 19 '11 at 17:49
@Prime, Are you looking for examples or an explanation? – Srivatsan Sep 19 '11 at 17:58

The intersection of all the open intervals centered at $0$ is just $\{0\}$, since $0$ is the only point that is a member of all of them.

But the intersection of all the open intervals whose lower boundary is $0$ is empty. (After all, what point could be a member all of them?) And they are nested, in that for any two of them, one is a subset of the other.

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