Can someone help me with this problem?

Consider the fixed point iteration

, k=0,1,2,...where

a) Show that the iteration converges for any initial guess

satisfying

b) Assume that the iteration converges, find m such that the method converges quadratically.

I think I got part a) by letting

, then take the derivative

. I then show that there exists

such that

, and by a theorem this implies the iteration converges for any initial guess in the specified interval.

For part b), I need to find m such that

exists, but I don't get anywhere from here. Hope someone can give a hand.