This problem is way in the wrong forum. It belongs in the University Math Help/Analysis, Topology and Differential Geometry forum.
I don't buy the result. You're dealing with fixed points here ( ). The problem is, there's no constraints on , other than the iteration scheme you've set up is known to converge, and there's no constraint on , either. So, I could easily produce a function that has two fixed points and . I then could pick The resulting sequence is convergent (it converges immediately!), but does not converge to .