Let $f$ be the function defined on the interval $] - \frac { 1 } { 3 } ; + \infty [$ by:
$$f ( x ) = \frac { 4 x } { 1 + 3 x }$$
We consider the sequence $(u _ { n })$ defined by: $u _ { 0 } = \frac { 1 } { 2 }$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$.

\begin{enumerate}
  \item Calculate $u _ { 1 }$.
  \item We admit that the function $f$ is increasing on the interval $] - \frac { 1 } { 3 } ; + \infty [$.\\
a. Show by induction that, for every natural number $n$, we have: $\frac { 1 } { 2 } \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 2$.\\
b. Deduce that the sequence $(u _ { n })$ is convergent.\\
c. We call $\ell$ the limit of the sequence $(u _ { n })$. Determine the value of $\ell$.
  \item a. Copy and complete the Python function below which, for every positive real number $E$, determines the smallest value $P$ such that: $1 - u _ { P } < E$.
\begin{verbatim}
def seuil(E) :
    u=0.5
    n = 0
    while
        u =
        n = n + 1
    return n
\end{verbatim}
b. Give the value returned by this program in the case where $E = 10 ^ { - 4 }$.
  \item We consider the sequence $(v _ { n })$ defined, for every natural number $n$, by:
$$v _ { n } = \frac { u _ { n } } { 1 - u _ { n } }$$
a. Show that the sequence $(v _ { n })$ is geometric with common ratio 4. Deduce, for every natural number $n$, the expression of $v _ { n }$ as a function of $n$.\\
b. Prove that, for every natural number $n$, we have: $u _ { n } = \frac { v _ { n } } { v _ { n } + 1 }$.\\
c. Show then that, for every natural number $n$, we have:
$$u _ { n } = \frac { 1 } { 1 + 0.25 ^ { n } }$$
Find by calculation the limit of the sequence $(u _ { n })$.
\end{enumerate}