Limit set of an infinite normal Kleinian subgroup ($N◃\Gamma ,|N|=\infty ⟹\Lambda (N)=\Lambda (\Gamma )$)

The Kleinian limit set of an infinite subgroup is nonempty. If it is normal, its limit set is invariant under the ambient group by base-point independence and conjugation. Minimality of the Kleinian limit set gives one inclusion, and subgroup containment gives the other. Neither properness nor finite generation of the subgroup is needed.