Sequence of Imaginary Reciprocals

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

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Point Sets: $45$