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I am stuck on the last problem of my complex variable homework.

The problem is given the set $S=\{\frac{i}{n} \mid n \text{ is an integer}\}$,

a.) List the points in $S$.

b.) What are the accumulation points of $S$?

So far I have the points being $i$, $i/2$, $i/3,\ldots , i/n$ but that seems like it is too easy for this class.

Am I missing something?

The second part, I would think there are not any accumulation points because none of the neighborhoods of the points will include any other points for a small neighborhood.

Any help would be appreciated.

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    Joe, that looks good to me.2011-10-11

1 Answers 1

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For every nonzero complex number $z$, let $A_z=\left\{\frac{z}n; n\geqslant1\right\}$. Then, for every positive $\varepsilon$:

  1. The set $A_z\cap D_\varepsilon$ is infinite,
  2. The set $A_z\setminus D_\varepsilon$ is finite.

Here, $D_\varepsilon=\{u\in\mathbb C;|u|<\varepsilon\}$. This shows that $A_z$ has one and only one accumulation point, which is $0$, for every $z\ne0$ and for example $z=i$.

To prove 1. and 2., note that for every nonzero $z$ and every positive $\varepsilon$, there exists a finite nonnegative integer $k$ such that $k\varepsilon\leqslant |z|<(k+1)\varepsilon$. Hence, $A_z\cap D_\varepsilon=\left\{\frac{z}n;n\geqslant k+1\right\}$ is always infinite and $A_z\setminus D_\varepsilon=\left\{\frac{z}n;1\leqslant n\leqslant k\right\}$ is always finite.

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    Right. Then the proof is valid.2011-10-11