$\displaystyle \mathbb{D}_f$ is the domain of $\displaystyle f$, and $\displaystyle \mathbb{R}_f$ is the range of $\displaystyle f$.

Let $\displaystyle f$ be some function, and let $\displaystyle x \in \mathbb{D}_f \cap \mathbb{R}_f$ satisfy $\displaystyle f(x) = x$. Let a series be defined by :

$\displaystyle \left \{

\begin{array}{l}

u_0 \in \mathbb{D}_f \\

u_{n + 1} = f(u_n)

\end{array}

\right.$

Show that if this series converges towards a finite value, then it converges towards $\displaystyle x$.

Note : you may use $\displaystyle f(x) = \frac{1}{x} + 1$, with $\displaystyle x = \phi$ as an example.

This can be explained because then $\displaystyle \frac{1}{x} + 1 = x$, yielding $\displaystyle x^2 - x - 1 = 0$.