Consider the sequence $(u_n)$ defined by:
$$u_0 = 8 \text{ and for every natural number } n,\quad u_{n+1} = u_n - \ln\left(\frac{u_n}{4}\right).$$

\begin{enumerate}
  \item a. Give the values rounded to the nearest hundredth of $u_1$ and $u_2$.\\
  b. Consider the mystery function defined below in Python. We admit that, for every strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$.
\begin{verbatim}
def mystery(k) :
    u = 8
    S = 0
    for i in range(k) :
        S = S + u
        u = u - log( u / 4)
    return S
\end{verbatim}
  The execution of \texttt{mystery(10)} returns 58.44045206721732. What does this result represent?\\
  c. Modify the previous function so that it returns the average of the first $k$ terms of the sequence $(u_n)$.

  \item Consider the function $f$ defined and differentiable on $]0; +\infty[$ by:
  $$f(x) = x - \ln\left(\frac{x}{4}\right).$$
  Study the variations of $f$ on $]0; +\infty[$ and draw its variation table. The exact value of the minimum of $f$ on $]0; +\infty[$ will be specified. Limits are not required.

  In the rest of the exercise, it will be noted that for every natural number $n$, $u_{n+1} = f(u_n)$.

  \item a. Prove, by induction, that for every natural number $n$, we have:
  $$1 \leqslant u_{n+1} \leqslant u_n.$$
  b. Deduce that the sequence $(u_n)$ converges to a real limit.

  Let $\ell$ denote the value of this limit.\\
  c. Solve the equation $f(x) = x$.\\
  d. Deduce the value of $\ell$.
\end{enumerate}