We consider the function $g$ defined on the interval $[0; 1]$ by
$$g(x) = 2x - x^2$$

\begin{enumerate}
  \item Show that the function $g$ is strictly increasing on the interval $[0; 1]$ and specify the values of $g(0)$ and $g(1)$.
\end{enumerate}

We consider the sequence $\left(u_n\right)$ defined by $\left\{\begin{array}{l} u_0 = \dfrac{1}{2} \\ u_{n+1} = g\left(u_n\right) \end{array}\right.$ for every natural number $n$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Calculate $u_1$ and $u_2$.
  \item Prove by induction that, for every natural number $n$, we have: $0 < u_n < u_{n+1} < 1$.
  \item Deduce that the sequence $(u_n)$ is convergent.
  \item Determine the limit $\ell$ of the sequence $(u_n)$.
\end{enumerate}

We consider the sequence $(\nu_n)$ defined for every natural number $n$ by $\nu_n = \ln\left(1 - u_n\right)$.

\begin{enumerate}
  \setcounter{enumi}{5}
  \item Prove that the sequence $\left(v_n\right)$ is a geometric sequence with common ratio 2 and specify its first term.
  \item Deduce an expression for $v_n$ as a function of $n$.
  \item Deduce an expression for $u_n$ as a function of $n$ and find again the limit determined in question 5.
  \item Copy and complete the Python script below so that it returns the rank $n$ from which the sequence exceeds 0.95.
\begin{verbatim}
def seuil() :
    n=0
    u=0.5
    while u < 0.95:
        n=...
        u=...
    return n
\end{verbatim}
\end{enumerate}