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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?

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    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
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    Yes, it would mean that. Because $x\leq y/n$ holds if and only if $nx\leq y$.2011-06-20
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    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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Nothing wrong with the proof by contradiction, but just to be non-contrary let me give a direct proof that the archimedean property implies the intersection is empty.

Note that $\bigcap\limits_{n=1}^{\infty}(0,y/n]\subseteq (0,y]$. Therefore, $$\bigcap_{n=1}^{\infty}(0,y/n] = \left(\cap_{n=1}^{\infty}(0,y/n]\right)\cap(0,y].$$ Now let $x\in (0,y]$. By the Archimedean property, since $0\lt x$ and $0\lt y$, there exists $k\in\mathbb{N}$ such that $kx\gt y$. Therefore, $x\gt y/k$, so $x\notin (0,y/k]$. Since $\bigcap\limits_{n=1}^{\infty}(0,y/n]\subseteq (0,y/k]$, we conclude that $x\notin \bigcap\limits_{n=1}^{\infty}(0,y/n]$.

That is: for every $x$, $x\in (0,y]$ implies $x\notin \bigcap\limits_{n=1}^{\infty}(0,y/n]$. Therefore, $$\left(\bigcap_{n=1}^{\infty}(0,y/n]\right)\cap (0,y] = \emptyset.$$ Thus, $$\bigcap_{n=1}^{\infty}(0,y/n] = \left(\bigcap_{n=1}^{\infty}(0,y/n]\right)\cap(0,y] = \emptyset,$$ proving the intersection is empty.

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    @Zev: Thanks for the assist.2011-06-21
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    No problem!2011-06-21
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    @Theo: Yes: I forgot that `\cap` does not automatically turn into an operator under suitable circumstances (the way that `-` does) and has to be manually made into one; and I navigated away instead of checking.2011-06-21
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    Oh, so disappointing... I thought I could tell you didn't know about LaTeX :)2011-06-21
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    @Theo: Plenty of that around, trust me; especially since I learned PlainTeX by reading *The LaTeX Book* (multiple times, as required) and stick to what works. Took me forever to switch to `align` from `eqnarray` and `array`...2011-06-21
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Indeed if $x\in\bigcap_{n=1}^\infty (0,y/n]$ then for every $n\in\mathbb N$ we have that $0

That is for every $n\in\mathbb N$ we have $0real number is smaller than only finitely many $y/n$.

The $x$ as above does not have this property, and indeed it will be a non-Archimedean infinitesimal number.