The fixed point is at  1/(1-m), and the vertical portions of the 
red line segments demonstrates that
1 + m + m^2 + ... = 1/(1-m).

m = 0.6 (slider from -0.95 to 0.95, step 0.01)
f(x) = 1.0 + m * x
g(x) = x
P = Intersect(f, g)
a = Segment((0.0, 0.0), (0.0, 1.0))
list1 = Sequence(Segment[(Iteration[f(x), 0.0, n], Iteration[f(x), 0.0, n + 1.0]), (Iteration[f(x), 0.0, n + 1.0], Iteration[f(x), 0.0, n + 1.0])], n, 0.0, 100.0)
list2 = Sequence(Segment[(Iteration[f(x), 0.0, n + 1.0], Iteration[f(x), 0.0, n + 1.0]), (Iteration[f(x), 0.0, n + 1.0], Iteration[f(x), 0.0, n + 2.0])], n, 0.0, 100.0)