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The cantor's intersection theorem states that if $X$ is a complete metric space, then, A decreasing nested sequence of non-empty compact subsets of has a non-empty intersection.

Consider the nested sets : $[-r,r]$ where $r \in (0,1]$. Then , clearly, $[-r,r]$ forms a decreasing diameter nested sequence of sets but whose intersection is empty as $r$ doesn't take the value $0$. Does this contradict the cantor's intersection theorem? Where could I be wrong? Thanks!

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We still have $0\in\displaystyle\bigcap_{r\in(0,1]}[-r,r]$ because for each $r\in(0,1]$, it is true that $0\in[-r,r]$.

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  • $\begingroup$ Oops sorry that was lame of me. Thanks $\endgroup$ – MathMan Apr 22 '18 at 7:17

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