Take $r=0$. Let $G=\langle a_1\rangle \times ...\times \langle a_n\rangle, \langle a_i\rangle$ infinite cyclic. $G$ is generated by $n$ elements and all relations in $A$ are relations in $G$. Therefor by von Dyck's theorem there is an epimorphism from $A$ to $G$. But $G$ is infinite and so, $A$ is infinite too.
For the case $r>0$ I have been unable to find a proof. I don't want a solution. I only ask for a hint which allows me to begin working in the problem.
Hint: $A$ is an abelian group, so think of it as a $\Bbb Z$-module. $A$ can be presented as the quotient of $\Bbb Z^n$ by the (free) submodule generated by the $r$ relations.