Sequence of Imaginary Reciprocals/Boundedness

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 bounded in $\C$.

Proof

Let $z \in S$.

Then, for example:

$\cmod z \le 2$

That is, $S$ is contained entirely within a circle of radius $2$ whose center is at the origin.

$\blacksquare$

Sources

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