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So here is the problem,

$\bigcap_{i=1}^\infty (0,\frac1i]$

personly I think the answer is either $0$ or $0^+$.

The answer is supposedly $\emptyset$.

I came here because I can't seem to prove it to my self that it's the empty set. I also couldn't find any info on this particular problem only problems like it Intersection of sets within space [0,1/i] vs (0,1/i) as i approaches infinity.

Help would be appreciated.

lefteh
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    What exactly is $0^+$? Notice that $0$ isn't actually in any of this family of sets. –  Oct 04 '18 at 22:21
  • I see $0^+$ as a value approaching $0$ from the positive side but not zero, at least that's my thought. – lefteh Oct 04 '18 at 22:25
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    There are no infinitesimals in $\mathbb{R}$, so there is no minimum positive number. So there is no $0^+$. –  Oct 04 '18 at 22:27
  • For 0 to be in the intersection, it would have to be in every one of the intervals $\left(0, \frac1n\right]$. But it isn't in any of them! – MJD Oct 05 '18 at 04:20

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Note that each set only contains positive numbers, so the intersection must also only contain positive numbers.

Now suppose we conjecture that $k>0$ is in the intersection. By the archimedian property, we can find $n\in\mathbb N$ such that $n>\frac1k$. This implies $\frac1n<k$, so $k\notin(0,\frac1n]$, a contradiction.

Hence the intersection is empty.

Rushabh Mehta
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