Finite implies finitely generated

This article gives the statement, and possibly proof, of a basic fact in group theory.
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Definition

Suppose $G$ is a finite group. Then $G$ is a finitely generated group.

Proof

If $|G|=n$ with $g_{1},\dots ,g_{n}$ its elements, then $G=\langle g_{1},\dots ,g_{n}\rangle$ follows from the closure of the binary operation on $G$.

Remark

Note the constructed generating set for $G$ in this proof may not be optimal. For example, the cyclic group of order $n$, for any $n$, can be generated by just $1$ element.