Equivalence of Definitions of Order of Group Element

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

Let $a \in G$ have finite order such that $\left|{a}\right| = k$.

Then:
 * $\left|{\left \langle {a} \right \rangle}\right| = k$

where $\left \langle {a} \right \rangle$ is the smallest subgroup of $G$ containing $a$.

That is, the order of the subgroup generated by $a$ is equal to the order of $a$.

(Some sources use this as the definition of the order of an element, and from it derive Order of an Element.)

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