Consider the function $f$ defined on $\mathbb{R}$ by
$$f(x) = \frac{3}{4}x^2 - 2x + 3$$
\begin{enumerate}
  \item Draw the table of variations of $f$ on $\mathbb{R}$.
  \item Deduce that for all $x$ belonging to the interval $\left[\frac{4}{3}; 2\right]$, $f(x)$ belongs to the interval $\left[\frac{4}{3}; 2\right]$.
  \item Prove that for all real $x$, $x \leq f(x)$.\\
  For this, one may prove that for all real $x$:
  $$f(x) - x = \frac{3}{4}(x - 2)^2.$$
\end{enumerate}
Consider the sequence $(u_n)$ defined by a real $u_0$ and for all natural integer $n$:
$$u_{n+1} = f(u_n).$$
We have therefore, for all natural integer $n$,
$$u_{n+1} = \frac{3}{4}u_n^2 - 2u_n + 3.$$
\begin{enumerate}
  \setcounter{enumi}{3}
  \item Study of the case: $\frac{4}{3} \leq u_0 \leq 2$.\\
  a. Prove by induction that, for all natural integer $n$,
  $$u_n \leq u_{n+1} \leq 2.$$
  b. Deduce that the sequence $(u_n)$ is convergent.\\
  c. Prove that the limit of the sequence is equal to 2.
  \item Study of the particular case: $u_0 = 3$.\\
  It is admitted that in this case the sequence $(u_n)$ tends to $+\infty$.\\
  Copy and complete the following ``threshold'' function written in Python, so that it returns the smallest value of $n$ such that $u_n$ is greater than or equal to 100.
\begin{verbatim}
def seuil() :
    u = 3
    n = 0
    while ...
        u = ...
        n = ...
    return n
\end{verbatim}
  \item Study of the case: $u_0 > 2$.\\
  Using a proof by contradiction, show that $(u_n)$ is not convergent.
\end{enumerate}