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Show that for every real number $y>0$, $$\bigcap_{n=1}^{\infty} (0, y/n] = \emptyset$$

So this would mean that $0< x \leq y/n$ for every positive integer $n$ which contradicts the Archimdean property?

  • 0
    By "this" you mean "$x\in\bigcap_{n=1}^{\infty} (0, y/n]$". This could be written more clearly, but the answer is yes.2011-06-20
  • 1
    Yes, it would mean that. Because $x\leq y/n$ holds if and only if $nx\leq y$.2011-06-20
  • 2
    Very good thinking, perfectly correct. Depending on what tools you have by now, it may have been intended that you first use the "nested interval" property to show that $\bigcap[0,y/n]$ only contains the point $0$.2011-06-20

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