# Proof of Newton – Fixed point iteration to converge to root of function

How to prove this inequality such that there exists a point between $$x_k$$ and $$r$$ such that:
$$x_{k+1} – r le frac{|f'(x_0) – f'(r)| + |f'(r) – f'(xi)|}{f'(x_0)} x_{k} – r$$

knowing that: $$x_{k+1} = x_k – frac{f(x_k)}{{f'(x_0)}}$$

where r is the root of the $$f(r) = 0$$

I know that this is newton method per Fixed point iteration one can find our local convergence. I tried the mean value theorem but I couldn’t formulate it in such way. But I can’t prove the existence of $$xi$$. Any pointouts would be of great help.