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

.