We consider the numerical sequence $(u_n)$ defined by its first term $u_0 = 2$ and for every natural number $n$, by:
$$u_{n+1} = \frac{2u_n + 1}{u_n + 2}$$
We admit that the sequence $(u_n)$ is well defined.

\begin{enumerate}
  \item Calculate the term $u_1$.
  \item We define the sequence $(a_n)$ for every natural number $n$, by:
$$a_n = \frac{u_n}{u_n - 1}$$
We admit that the sequence $(a_n)$ is well defined.\\
a. Calculate $a_0$ and $a_1$.\\
b. Prove that, for every natural number $n$, $a_{n+1} = 3a_n - 1$.\\
c. Prove by induction that, for every natural number $n$ greater than or equal to 1,
$$a_n \geqslant 3n - 1$$
d. Deduce the limit of the sequence $(a_n)$.
  \item We wish to study the limit of the sequence $(u_n)$.\\
a. Prove that for every natural number $n$, $u_n = \frac{a_n}{a_n - 1}$.\\
b. Deduce the limit of the sequence $(u_n)$.
  \item We admit that the sequence $(u_n)$ is decreasing.

We consider the following program written in Python:
\begin{verbatim}
def algo(p):
    u=2
    n=0
    while u-1>p:
        u=(2*u+1)/(u+2)
        n=n+1
    return (n,u)
\end{verbatim}
a. Interpret the values $n$ and u returned by the call to the function algo(p) in the context of the exercise.\\
b. Give, without justification, the value of $n$ for $p = 0.001$.
\end{enumerate}