This is a comment on Pietro Majer's answer, with graphics. The first figure shows $x^2$ (red) and the first five functions of Pietro's method approaching it (blue). The second figure shows the double derivatives thereof. As you can see, the functions are converging but their second derivatives are not.

[![enter image description here][1]][1]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/uoe6e.png
  [2]: https://i.sstatic.net/y12EP.png