# 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

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