Consider the function $f$ defined on $]-1.5; +\infty[$ by
$$f(x) = \ln(2x + 3) - 1$$
The purpose of this exercise is to study the convergence of the sequence $(u_{n})$ defined by:
$$u_{0} = 0 \text{ and } u_{n+1} = f(u_{n}) \text{ for all natural integer } n.$$

\textbf{Part A: Study of an auxiliary function}\\
Consider the function $g$ defined on $]-1.5; +\infty[$ by $g(x) = f(x) - x$.

\begin{enumerate}
  \item Determine the limit of the function $g$ at $-1.5$.
\end{enumerate}

We admit that the limit of the function $g$ at $+\infty$ is $-\infty$.

\begin{enumerate}
  \setcounter{enumi}{1}
  \item Study the variations of the function $g$ on $]-1.5; +\infty[$.
  \item a. Prove that, in the interval $]-0.5; +\infty[$, the equation $g(x) = 0$ admits a unique solution $\alpha$.\\
  b. Determine an interval containing $\alpha$ with amplitude $10^{-2}$.
\end{enumerate}

\textbf{Part B: Study of the sequence $(u_{n})$}\\
We admit that the function $f$ is strictly increasing on $]-1.5; +\infty[$.

\begin{enumerate}
  \item Let $x$ be a real number. Show that if $x \in [-1; \alpha]$ then $f(x) \in [-1; \alpha]$.
  \item a. Prove by induction that for all natural integer $n$:
$$-1 \leqslant u_{n} \leqslant u_{n+1} \leqslant \alpha.$$
  b. Deduce that the sequence $(u_{n})$ converges.
\end{enumerate}