Let $G$ be a group, $a\in G$. Show that $C(a)=G \iff a \in Z(G)$ where $C(a)$ denotes the centralizer of the element $a$ and $Z(G)$ denotes the center of the group.
My proof:
Assume $a \in Z(G)$. Then, $ax=xa, \forall x\in G$ by definition. Since all elements of $G$ commute with $a$, all elements of $G$ are in $C(a)$. Hence, $G\subseteq C(a)$ as sets. Moreover, $C(a) \subseteq G$ by definition. Therefore, $C(a)=G$.
Assume $C(a)=G$. Then, $a$ commutes with every element of $G$. By definition, $a\in Z(G)$.
And the proof is finished. Is this too simple? I feel like this proof is almost trivial.