Sequence of Imaginary Reciprocals/Not Compact

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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 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