We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma)
of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We
show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are
(eventually) elementary or contain normal finite subgroups of arbitrarily high order, their
algebraic limit is a discrete nonelementary subgroup of G. In the case of divergent
sequences (Gamma_i) we show that the limiting action on a real tree T satisfies certain
semistability condition, which generalizes the notion of stability introduced by Rips. We
then verify that the group Gamma splits as an amalgam or HNN extension of finitely
generated groups, so that the edge group has an amenable image in the isometry group of T.