A recurrence turns each output into the next input.
A fixed point satisfies L=rL+1. Subtracting this equation from the recurrence gives xₙ−L=rⁿ(x₀−L). Thus |r|<1 guarantees convergence. Negative r alternates around the fixed point; positive r approaches from one side in this affine model.
A fixed point need not attract nearby iterates when |r|≥1; that range is outside this experiment.