Sequence of Imaginary Reciprocals/Not Compact

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 set $S$ is not compact.

Proof

From Sequence of Imaginary Reciprocals: Boundedness, $S$ is bounded in $\C$.

But from Sequence of Imaginary Reciprocals: Closedness, $S$ is not closed.

Hence the result by definition of compact.

$\blacksquare$

Sources

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