A group H is called finitely generated if there is a finite set A s.t. $\displaystyle H=<A>$

Prove that every finite group is finitely generated.

$\displaystyle <A>=\bigcap_{\begin{cases}A\subset H\\H\leq G\end{cases}}H$

I don't know how to set that up with out the { and make it smaller under the intersection.

Anyways. Can I just say the intersection of finite elements is always finite? How can this be shown?