Any help with the following:
Problem: Consider the fixed point problem: $x=f(x)$ and given: $x_{n+1}=\frac{n}{n+1}f\left ( x_{n} \right )$. If $x_{0}$ is a fixed point where \left | f^{'}\left ( x_{0} \right ) \right |< 1, prove convergence.
Now consider the case \left | f^{'}\left ( x_{0} \right ) \right |=1. In some cases this iteration converges then. Formulate such a theorem and prove it.
For the first part:here is what I did: $x_{n+1}=g(x_{n},n)=\frac{n}{n+1}g\left ( x_{n} \right )$ and then \left | g^{'}(x_{0},n) \right |=\left | \frac{n}{n+1}g^{'}\left ( x_{0} \right ) \right |< 1 since \left | f^{'}\left ( x_{0} \right ) \right |< 1 and $\frac{n}{n+1}< 1$. So, the convergence follows from the contraction mapping theorem.
For the second part, I don't have any idea at all. Any help please with the second part? and also do let me know if what I did in the first part is correct or not.