Separable Discrete Space is Countable/Proof 1

Proof
Let $T$ be separable.

By Space is Separable iff Density not greater than Aleph Zero:
 * $\map d T \le \aleph_0$

where:
 * $\map d T$ denotes the density of $T$
 * $\aleph$ denotes the aleph mapping.

By definition of density:
 * $\exists A \subseteq S: A$ is everywhere dense and $\map d T = \card A$

where $\card A$ denotes the cardinality of $A$.

By definition of everywhere dense set:
 * $A^- = S$

where $A^-$ denotes the closure of $A$.

By Set in Discrete Topology is Clopen:
 * $A$ is closed

Then by Set is Closed iff Equals Topological Closure:
 * $A^- = A$

Thus by Countable iff Cardinality not greater than Aleph Zero:
 * $S$ is countable.