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Sorry for the non descriptive title.

I have an assignment about the limit point of $A = \{1/n:n \in \mathbb{N}\}$

To show that $0$ is the only limit point of $A$ I assumed if $z \in (0,1] - A$, then $z$ is between two points in $A$, and showed that there's an open sphere around $z$ that is disjoint from $A$.

It seems obvious that $z$ is between points in $A$, but I got a note on my assignment that I need to show that $z$ is between points in $A$ because it is not obvious.

I know that $z \neq 0$ (because it's not in the range), $z$ cannot equal any point in $A$, so it seems like every other point in $(0,1]$ is between points in $A$.

How do I show this?

Matthew Riley
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    I'm guessing that what your grader had in mind was something to the effect of "there is an $n$ such that $1/n > z$ and $1/(n+1) < z$", which you might prove using the Archimedian property of the real numbers. – Ben Grossmann Sep 28 '18 at 17:10

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Suppose $z\in (0,1]\setminus A$. Since $z\notin A$ and $z\not=0$, we know that $1/z$ is a well-defined number that cannot be an integer. Let $n=[1/z]$ (the floor of $1/z$). Then, $n<1/z<n+1$. Hence, $1/(n+1)<z<1/n$.

Let $r=\min\{z-1/(n+1),1/n-z\}$. Then, $B_{r}(z)$ is disjoint from $A$.

ervx
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