Equivalence of Definitions of Order of Group Element

Theorem
Let $G$ be a group whose identity is $e$.

Let $x \in G$.

The following definitions of the order of $a$ in $G$ are equivalent:

Proof
It follows straight away from List of Elements in Finite Cyclic Group that $\left|{\left \langle {a} \right \rangle}\right| = k$:
 * $\left \langle {a} \right \rangle = \left\{{a^0, a^1, a^2, \ldots, a^{k - 1}}\right\}$