We consider the sequences $(u_{n})$ and $(v_{n})$ defined for every natural integer $n$ by:

$$\left\{ \begin{array}{l} u_{0} = v_{0} = 1 \\ u_{n+1} = u_{n} + v_{n} \\ v_{n+1} = 2u_{n} + v_{n} \end{array} \right.$$

Throughout the rest of the exercise, we assume that the sequences $(u_{n})$ and $(v_{n})$ are strictly positive.

\begin{enumerate}
  \item a. Calculate $u_{1}$ and $v_{1}$.\\
b. Prove that the sequence $(v_{n})$ is strictly increasing, then deduce that for every natural integer $n$, $v_{n} \geqslant 1$.\\
c. Prove by induction that for every natural integer $n$, we have: $u_{n} \geqslant n + 1$.\\
d. Deduce the limit of the sequence $(u_{n})$.
  \item We set, for every natural integer $n$:
$$r_{n} = \frac{v_{n}}{u_{n}}.$$
We assume that:
$$r_{n}^{2} = 2 + \frac{(-1)^{n+1}}{u_{n}^{2}}$$
a. Prove that for every natural integer $n$:
$$-\frac{1}{u_{n}^{2}} \leqslant \frac{(-1)^{n+1}}{u_{n}^{2}} \leqslant \frac{1}{u_{n}^{2}}.$$
b. Deduce:
$$\lim_{n \rightarrow +\infty} \frac{(-1)^{n+1}}{u_{n}^{2}}$$
c. Determine the limit of the sequence $\left(r_{n}^{2}\right)$ and deduce that $\left(r_{n}\right)$ converges to $\sqrt{2}$.\\
d. Prove that for every natural integer $n$,
$$r_{n+1} = \frac{2 + r_{n}}{1 + r_{n}}$$
e. Consider the following program written in Python language:
\begin{verbatim}
def seuil() :
    n = 0
    r = l
    while abs(r-sqrt(2)) > 10**(-4) :
        r = (2+r)/(1+r)
        n = n+1
    return n
\end{verbatim}
(abs denotes absolute value, sqrt the square root and $10^{**}(-4)$ represents $10^{-4}$). The value of $n$ returned by this program is 5. What does it correspond to?
\end{enumerate}