Nilpotent implies no proper contranormal subgroup

Statement
In a nilpotent group, there is no proper  contranormal subgroup. A contranormal subgroup is a subgroup whose normal closure is the whole group.

Facts used

 * 1) uses::Nilpotent implies every subgroup is subnormal

Proof
The proof follows directly from Fact (1), and the observation that a proper subnormal subgroup cannot be contranormal because of the definition of subnormal.