Sequence of Imaginary Reciprocals/Closure
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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 closure of $S$ is the set:
- $\set {0, i, \dfrac i 2, \dfrac i 3, \dfrac i 3, \ldots}$
Proof
By definition, the closure of $S$ is the set $S$ along with all its limit points.
Sequence of Imaginary Reciprocals: Limit Points, $S$ has one limit point $z = 0$.
Hence the result.
$\blacksquare$
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Point Sets: $45 \ \text {(h)}$