Sequence of Imaginary Reciprocals/Closure

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Theorem

Consider the subset $S$ of the complex plane defined as:

$S := \set {\dfrac i n : n \in \Z_{>0} }$

That is:

$S := \set {i, \dfrac i 2, \dfrac i 3, \dfrac i 4, \ldots}$

where $i$ is the imaginary unit.


The closure of $S$ is the set:

$\set {0, i, \dfrac i 2, \dfrac i 3, \dfrac i 3, \ldots}$


Proof

By definition, the closure of $S$ is the set $S$ along with all its limit points.

Sequence of Imaginary Reciprocals: Limit Points, $S$ has one limit point $z = 0$.

Hence the result.

$\blacksquare$


Sources