We consider sequences of discrete subgroups Γ i = ρ i (Γ) of a rank 1 Lie group G, with Γ finitely generated. We show that, for algebraically convergent sequences (Γ i ), unless Γ 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 (Γ i ) we show that the resulting action Γ T on a real tree satisfies certain semistability condition, which generalizes the notion of stability introduced by Rips. We then verify that the group Γ splits as an amalgam or HNN extension of finitely generated groups, so that the edge group has an amenable image in Isom(T ).