Sufficiency of r = m for proper convergence of candidate sequences

Establish whether, for a smooth contraction f: I → I with fixed point L and derivative m = |f'(L)| in (0,1), choosing r = m is sufficient to ensure that the sequence c_n = |L − f^{(n)}(t_0)| / r^n converges to a finite positive limit for all initial values t_0 ∈ I \ {L}.

Background

The paper introduces the generalized sequence c_n = (L − f^{(n)}(t_0)) / r^n and shows that r must equal m = |f'(L)| for c_n to have a finite positive limit. However, while necessity is established via bounds derived from the mean value theorem, the authors do not determine whether r = m guarantees convergence to a positive finite limit under the stated hypotheses.

This question concerns the exact conditions under which the residual sequence obtained by dividing out the exponential decay rate m^n converges properly for arbitrary smooth contractions with fixed point L.