\textbf{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}

\textbf{Part B: application to the study of a sequence}

We set $u _ { 0 } = 1$ and for every natural number $n , u _ { n + 1 } = u _ { n } - \ln \left( 1 + u _ { n } \right)$. 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 every natural number $n , \quad u _ { n } \geqslant 0$.\\
b. Prove that the sequence $( u _ { n } )$ is decreasing, and deduce that for every natural number $n , \quad u _ { n } \leqslant 1$.\\
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}