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I'm not sure if this is true or not. I find it very intuitive:

$\bigcap_{k=1}^{\infty} [n,n+\frac{1}{n^{2}}+\frac{1}{k}) = [n,n+\frac{1}{n^{2}}] $

Is this true and how do I then construct a proof? If the case is that the above statement is false, are there any other ways to express $[n,n+\frac{1}{n^{2}}]$?

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up vote 2 down vote accepted

Since $n+1/n^2+\frac 1k>n+1/n^2$, $\supset$ is true. If $n\leq x< n+1/n^2+1/k$ for all integer $k$, then $x\leq n+1/n^2$. Indeed, if we had $x>n+1/n^2$, then $x=n+1/n^2+\delta$ where $\delta>0$. Now take $k$ such that $\frac 1k<\delta$ to get a contradiction.

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@AlexBecker Right, fixed now. – Davide Giraudo Sep 10 '12 at 20:33

To construct a proof, you need to observe that all the points in $[n,n+\frac 1{n^2}]$ are in every one of the sets on the left, then show that any point outside $[n,n+\frac 1{n^2}]$ is not in (at least) one of the sets on the left. Points less than $n$ are no problem, so if I give you a point $x$, which is greater than $n+\frac 1{n^2}$, can you find a specific set on the left that doesn't include it?

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