\section*{Exercise 3 (5 points)}
For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.\\
The five questions in this exercise are independent.

\begin{enumerate}
  \item Consider a sequence ( $t _ { n }$ ) satisfying the recurrence relation:
$$\text { for all natural integer } n , t _ { n + 1 } = - 0.8 t _ { n } + 18 .$$
Statement 1: The sequence ( $w _ { n }$ ) defined for all natural integer $n$ by $w _ { n } = t _ { n } - 10$ is geometric.

  \item Consider a sequence ( $S _ { n }$ ) that satisfies for all non-zero natural integer $n$:
$$3 n - 4 \leqslant S _ { n } \leqslant 3 n + 4 .$$
The sequence ( $u _ { n }$ ) is defined, for all non-zero natural integer $n$, by: $u _ { n } = \frac { S _ { n } } { n }$.\\
Statement 2: The sequence ( $u _ { n }$ ) converges.

  \item Consider the sequence $\left( v _ { n } \right)$ defined by:
$$v _ { 1 } = 2 \text { and for all natural integer } n \geqslant 1 , v _ { n + 1 } = 2 - \frac { 1 } { v _ { n } } .$$
Statement 3: For all natural integer $n \geqslant 1 , v _ { n } = \frac { n + 1 } { n }$.

  \item Consider the sequence ( $u _ { n }$ ) defined for all natural integer $n$ by $u _ { n } = \mathrm { e } ^ { n } - n$.\\
Statement 4: The sequence $\left( u _ { n } \right)$ converges.

  \item Consider the sequence ( $u _ { n }$ ) defined using the script written below in Python language, which returns the value of $u _ { n }$.
\begin{verbatim}
def u(n) :
    valeur = 2
    for k in range(n) :
        valeur = 0.5 * (valeur + 2/valeur)
    return valeur
\end{verbatim}
We admit that ( $u _ { n }$ ) is decreasing and satisfies for all natural integer $n$:
$$\sqrt { 2 } \leqslant u _ { n } \leqslant 2 .$$
Statement 5: The sequence $\left( u _ { n } \right)$ converges to $\sqrt { 2 }$.
\end{enumerate}