  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.
