Let $\left(u_n\right)$ be the sequence defined by $u_0 = 1$ and for every natural integer $n$
$$u_{n+1} = \frac{u_n}{1 + u_n}$$

\begin{enumerate}
  \item a. Calculate the terms $u_1, u_2$ and $u_3$. Give the results as irreducible fractions.\\
b. Copy the Python script below and complete lines 3 and 6 so that \texttt{liste($k$)} takes as parameter a natural integer $k$ and returns the list of the first values of the sequence $\left(u_n\right)$ from $u_0$ to $u_k$.
\begin{verbatim}
def liste(k) :
    L = []
    u=...
    for i in range(0, k+1) :
        L.append(u)
        u = ...
    return(L)
\end{verbatim}

  \item It is admitted that, for every natural integer $n$, $u_n$ is strictly positive. Determine the direction of variation of the sequence $(u_n)$.
  \item Deduce that the sequence $(u_n)$ converges.
  \item Determine the value of its limit.
  \item a. Conjecture an expression of $u_n$ as a function of $n$.\\
b. Prove by induction the previous conjecture.
\end{enumerate}