Sequence of Imaginary Reciprocals/Closedness

Theorem
The set $S$ is not closed.

Proof
From Sequence of Imaginary Reciprocals: Limit Points, $S$ has one limit point $z = 0$.

But:
 * $\nexists n \in \N: \dfrac 1 n = 0$

so $0 \notin S$.

As $S$ does not contain (all) its limit point(s), it follows by definition that $S$ is not closed.