Prove that an Abelian group with two elements of order 2 must have a subgroup of order 4.

My proof:

Suppose that G is an Abelian group, and let $\displaystyle a,b \in G$ such that $\displaystyle <a> = 2, <b> = 2$, so we have $\displaystyle a^{2}=e, b^{2}=e$.

From a theroem, we know that <a> and <b> are subgroups of G.

Am I starting this right?