Finitely generated abelian implies residually finite

Statement
Any finitely generated abelian group (i.e., an abelian group that is also a finitely generated group) is a residually finite group -- for every non-identity element, there is a normal subgroup of finite index not containing it.

Related facts

 * Finitely generated not implies residually finite
 * Residually finite not implies finitely generated
 * Free implies residually finite
 * Free abelian implies residually finite
 * Residual finiteness is direct product-closed

Facts used

 * 1) Structure theorem for finitely generated abelian group
 * 2) The infinite cyclic group is residually finite.
 * 3) Residual finiteness is direct product-closed

Proof using given facts
By fact (1), any finitely generated abelian group is a direct product of copies of the infinite cyclic group and a finite group. The infinite cyclic group is residually finite (fact (2)) and the finite group is residually finite, so by fact (3), the whole group is residually finite.