Sequence of Imaginary Reciprocals

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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.


Then $S$ has the following properties:


Boundedness

The set $S$ is bounded in $\C$.


Limit Points

The set $S$ has exactly one limit point, and that is $z = 0$.


Closedness

The set $S$ is not closed.


Boundary Points

Every point of $S$, along with the point $z = 0$, is a boundary point of $S$.


Interior

No point of $S$ is an interior point.


Openness

$S$ is not an open set.


Connectedness

$S$ is not connected.


Not an Open Region

$S$ is not an open region.


Closure

The closure of $S$ is the set:

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


Complement

The complement of $S$ in $\C$ is the set:

$\C \setminus \set {i, \dfrac i 2, \dfrac i 3, \dfrac i 3, \ldots}$


Countability

The set $S$ is countably infinite.


Not Compact

The set $S$ is not compact.


Closure is Compact

The closure $S^-$ of the set $S$ is compact.


Sources