Sequence of Imaginary Reciprocals/Closedness
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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 closed.
Proof
From Sequence of Imaginary Reciprocals: Limit Points, $S$ has one limit point $z = 0$.
But:
- $\nexists n \in \N: \dfrac i n = 0$
so $0 \notin S$.
As $S$ does not contain (all) its limit point(s), it follows by definition that $S$ is not closed.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Point Sets: $45 \ \text {(c)}$