Consider the function $f$ defined for all real $x$ by:
$$f ( x ) = \ln \left( \mathrm { e } ^ { \frac { x } { 2 } } + 2 \right)$$
It is admitted that the function $f$ is differentiable on $\mathbb { R }$.\\
Consider the sequence $(u_n)$ defined by $u _ { 0 } = \ln ( 9 )$ and, for all natural integer $n$,
$$u _ { n + 1 } = f \left( u _ { n } \right)$$
\begin{enumerate}
  \item Show that the function $f$ is strictly increasing on $\mathbb { R }$.
  \item Show that $f ( 2 \ln ( 2 ) ) = 2 \ln ( 2 )$.
  \item Show that $u _ { 1 } = \ln ( 5 )$.
  \item Show by induction that for all natural integer $n$, we have:
$$2 \ln ( 2 ) \leqslant u _ { n + 1 } \leqslant u _ { n }$$
  \item Deduce that the sequence $(u_n)$ converges.
  \item a. Solve in $\mathbb { R }$ the equation $X ^ { 2 } - X - 2 = 0$.\\
b. Deduce the set of solutions on $\mathbb { R }$ of the equation:
$$\mathrm { e } ^ { x } - \mathrm { e } ^ { \frac { x } { 2 } } - 2 = 0$$
c. Deduce the set of solutions on $\mathbb { R }$ of the equation $f ( x ) = x$.\\
d. Determine the limit of the sequence $\left( u _ { n } \right)$.
\end{enumerate}