Central Subgroup is Normal/Proof 2

Theorem
Let $G$ be a group.

Let $H$ be a central subgroup of $G$.

Then $H$ is a normal subgroup of $G$.

Proof
Let $H$ be a central subgroup of $G$.

By definition of central subgroup:
 * $H \subseteq Z \left({G}\right)$

where $Z \left({G}\right)$ is the center of $G$.

Then: