Residual finiteness is not quotient-closed

Statement
A quotient group of a residually finite group need not be a residually finite group.

Facts used

 * 1) uses::Free implies residually finite
 * 2) uses::Every group is a quotient of a free group

A generic example
Any free group is a residually finite group (fact (1)). Every group is isomorphic to a quotient group of a free group (fact (2)). However, there do exist groups that are not residually finite -- for instance, infinite simple groups.

A specific example
Consider the $$p$$-quasicyclic group for any prime $$p$$. This is a quotient of a direct product of finite groups -- namely, the direct product of all cyclic groups of prime power order for the prime $$p$$. The latter is clearly residually finite. On the other hand, the quasicyclic group is not -- any quotient by a proper subgroup is isomorphic to the $$p$$-quasicyclic group itself.