Residually finite not implies finitely generated

Statement
A residually finite group need not be a finitely generated group. In particular, the property of being a fact about::finitely generated residually finite group is strictly stronger than the property of being a residually finite group.

Related facts

 * Finitely generated not implies residually finite
 * Finitely generated abelian implies residually finite
 * Finitely generated and residually finite implies Hopfian
 * Residually finite not implies Hopfian

Proof
Consider an external direct product (or a restricted external direct product) of infinitely many nontrivial finite groups. In neither case is it finitely generated -- in the unrestricted case because it is uncountable, and in the restricted case because it is a locally finite group that is infinite. However, in both cases, the group is residually finite, using the definition of residually finite as being a subdirect product of finite groups.