Center of Group is Subgroup/Proof 2

Theorem
The center $Z \left({G}\right)$ of any group $G$ is a subgroup of $G$.

Proof
We have the result Center is Intersection of Centralizers.

That is, $Z \left({G}\right)$ is the intersection of all the centralizers of $G$.

All of these are subgroups of $G$ by Centralizer of Group Element is Subgroup.

Thus from Intersection of Subgroups, $Z \left({G}\right)$ is also a subgroup of $G$.