Sequence of Imaginary Reciprocals/Connectedness

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.

$S$ is not connected.

Proof

Let $z_1 \in S$ and $z_2 \in S$ be joined by a polygonal path $P$.

Then there are points of $P$ which are not in $S$.

Hence, by definition, $S$ is not connected.

$\blacksquare$

Sources

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