Fixed point iteration exercise $x=g(x)=-2^x+4$

Question

In the question below, why is it that $|g′(\xi_{n})|≥2⋅ln(2)$ for all $n=1,2,3,...$ How do I know that $\xi_{n}\in [1,2]$ for $n=1,2,..$ ?

The handwritten answer below is provided by someone else.

Answer 1

This seems to be an argument by contradiction. Assume that the iteration sequence converged to the fixed point, so it would have to, at some point, come close to it and especially stay inside the given interval.

Then against that you would get this exploding lower bound on the distance to the fixed point, giving a contradiction to the assumption of convergence.