Let $(u_{n})$ be the sequence defined by $u_{0} = 1$ and, for every natural integer $n$, $u_{n+1} = u_{n} - \ln\left(u_{n}^{2} + 1\right)$.
Show by induction that, for every natural integer $n$, $u_{n}$ belongs to $[0;1]$.
Study the variations of the sequence $(u_{n})$.
Show that the sequence $(u_{n})$ is convergent.
We denote by $\ell$ its limit, and we admit that $\ell$ satisfies the equality $f(\ell) = \ell$. Deduce the value of $\ell$.
Let $(u_{n})$ be the sequence defined by $u_{0} = 1$ and, for every natural integer $n$, $u_{n+1} = u_{n} - \ln\left(u_{n}^{2} + 1\right)$.
\begin{enumerate}
\item Show by induction that, for every natural integer $n$, $u_{n}$ belongs to $[0;1]$.
\item Study the variations of the sequence $(u_{n})$.
\item Show that the sequence $(u_{n})$ is convergent.
\item We denote by $\ell$ its limit, and we admit that $\ell$ satisfies the equality $f(\ell) = \ell$. Deduce the value of $\ell$.
\end{enumerate}