Is that's all? Thank you. :-)
A group $H$ is called finitely generated if there is a finite set $A$ such that $H = \left \langle A \right \rangle$ . Prove that every finite group is finitely generated.
Because $\forall a \in A, a\in H$. Hence, if $A$ is infinite, $H$ is infinite. Hence every finite group is finitely generated.
Your proof is correct: If there are infinitely many distinct generators, there are infinitely many distinct group elements.
For an alternative (simpler?) proof, just note that $H = \langle H \rangle$.