Let $a$ be a fixed non-zero real number. The purpose of this exercise is to study the sequence $(u_n)$ defined by:
$$u_0 = a \quad \text{and, for all } n \text{ in } \mathbb{N}, \quad u_{n+1} = \mathrm{e}^{2u_n} - \mathrm{e}^{u_n}.$$
Note that this equality can also be written: $u_{n+1} = e^{u_n}(\mathrm{e}^{u_n} - 1)$.

\begin{enumerate}
  \item Let $g$ be the function defined for all real $x$ by
$$g(x) = \mathrm{e}^{2x} - \mathrm{e}^x - x.$$
a) Calculate $g'(x)$ and prove that, for all real $x$: $g'(x) = (\mathrm{e}^x - 1)(2\mathrm{e}^x + 1)$.\\
b) Determine the variations of the function $g$ and give the value of its minimum.\\
c) By noting that $u_{n+1} - u_n = g(u_n)$, study the direction of variation of the sequence $(u_n)$.

  \item In this question, we assume that $a \leqslant 0$.\\
a) Prove by induction that, for all natural integer $n$, $u_n \leqslant 0$.\\
b) Deduce from the previous questions that the sequence $(u_n)$ is convergent.\\
c) In the case where $a$ equals 0, give the limit of the sequence $(u_n)$.

  \item In this question, we assume that $a > 0$.

Since the sequence $(u_n)$ is increasing, question 1 allows us to assert that, for all natural integer $n$, $u_n \geqslant a$.\\
a) Prove that, for all natural integer $n$, we have: $u_{n+1} - u_n \geqslant g(a)$.\\
b) Prove by induction that, for all natural integer $n$, we have:
$$u_n \geqslant a + n \times g(a).$$
c) Determine the limit of the sequence $(u_n)$.

  \item In this question, we take $a = 0.02$.

The following algorithm is intended to determine the smallest integer $n$ such that $u_n > M$, where $M$ denotes a positive real number. This algorithm is incomplete.

\begin{center}
\begin{tabular}{|l|l|}
\hline
Variables & $n$ is an integer, $u$ and $M$ are two real numbers \\
\hline
Initialization & \begin{tabular}{l}
$u$ takes the value 0.02 \\
$n$ takes the value 0 \\
Enter the value of $M$ \\
\end{tabular} \\
\hline
Processing & While $\cdots$ \\
 & $\ldots$ \\
 & $\ldots$ \\
End while &  \\
\end{tabular}
\end{center}

a) On your paper, rewrite the ``Processing'' part by completing it.\\
b) Using a calculator, determine the value that this algorithm will display if $M = 60$.
\end{enumerate}