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.