Closed Set/Complex Analysis/Examples/Closed Unit Circle

Example of Closed Set in the context of Complex Analysis
Let $S$ be the subset of the complex plane defined as:
 * $\cmod z \le 1$

where $\cmod z$ denotes the complex modulus of $z$.

Then $S$ is closed.

Proof
By definition, $S$ is closed $S$ contains all its limit points.

$S$ is not closed.

Then $\exists z_1 \in \C: z_1 \notin S$ such that $z_1$ is a limit point of $S$.

We have that $z_1 \notin S$, so $\cmod {z_1} > 1$.

Let $\epsilon \in \R: \epsilon < \cmod {z_1} - 1$.

Then:
 * $\map {\N_\epsilon} {z_1} \cap S = \O$

and so $z_1$ is not a limit point of $S$ after all.

Thus there is no such $z_1$.

So by Proof by Contradiction, $S$ is closed.