\section*{Part A: establishing an inequality}
On the interval $[0; +\infty[$, we define the function $f$ by $f(x) = x - \ln(x+1)$.

\begin{enumerate}
  \item Study the monotonicity of the function $f$ on the interval $[0; +\infty[$.
  \item Deduce that for all $x \in [0; +\infty[,\; \ln(x+1) \leqslant x$.
\end{enumerate}

\section*{Part B: application to the study of a sequence}
We set $u_0 = 1$ and for all natural number $n$, $u_{n+1} = u_n - \ln(1 + u_n)$. We admit that the sequence with general term $u_n$ is well defined.

\begin{enumerate}
  \item Calculate an approximate value to $10^{-3}$ of $u_2$.
  \item a. Prove by induction that for all natural number $n$, $u_n \geqslant 0$.\\
b. Prove that the sequence $(u_n)$ is decreasing, and deduce that for all natural number $n$, $u_n \leqslant 1$.\\
c. Show that the sequence $(u_n)$ is convergent.
  \item Let $\ell$ denote the limit of the sequence $(u_n)$ and we admit that $\ell = f(\ell)$, where $f$ is the function defined in part A. Deduce the value of $\ell$.
  \item a. Write an algorithm which, for a given natural number $p$, allows us to determine the smallest rank $N$ from which all terms of the sequence $(u_n)$ are less than $10^{-p}$.\\
b. Determine the smallest natural number $n$ from which all terms of the sequence $(u_n)$ are less than $10^{-15}$.
\end{enumerate}