The objective of this exercise is to conjecture in Part A and then prove in Part B the behavior of a sequence.\\
The two parts can, however, be treated independently.\\
We consider the sequence ( $u _ { n }$ ) defined by $u _ { 0 } = 3$ and for all $n \in \mathbb { N }$ :

$$u _ { n + 1 } = \frac { 4 } { 5 - u _ { n } } .$$

\textbf{Part A}
\begin{enumerate}
  \item Copy and complete the following Python function suite(n) which takes as parameter the rank $n$ and returns the value of the term $u _ { n }$.
\end{enumerate}

\begin{verbatim}
def suite(n):
    u = ...
    for i in range(n) :
        ...
    return u
\end{verbatim}

\begin{enumerate}
  \setcounter{enumi}{1}
  \item The execution of suite(2) returns 1.3333333333333333 .
\end{enumerate}

Perform a calculation to verify and explain this output.\\
3. Using the outputs below, make a conjecture about the direction of variation and a conjecture about the convergence of the sequence ( $u _ { n }$ ).

\begin{verbatim}
> suite(2)
1.3333333333333333
>> suite(5)
1.0058479532163742
>> suite(10)
1.0000057220349845
> suite(20)
1.000000000005457
\end{verbatim}

\textbf{Part B}\\
We consider the function $f$ defined and differentiable on the interval $] - \infty ; 5 [$ by:

$$f ( x ) = \frac { 4 } { 5 - x }$$

Thus, the sequence ( $u _ { n }$ ) is defined by $u _ { 0 } = 3$ and for all $n \in \mathbb { N } , u _ { n + 1 } = f \left( u _ { n } \right)$.

\begin{enumerate}
  \item Show that the function $f$ is increasing on the interval $] - \infty$; 5[.
  \item Prove by induction that for every natural integer $n$ we have:
\end{enumerate}

$$1 \leqslant u _ { n + 1 } \leqslant u _ { n } \leqslant 4 .$$

\begin{enumerate}
  \setcounter{enumi}{2}
  \item a. Let $x$ be a real number in the interval $] - \infty$; 5[.\\
Prove the following equivalence:

$$f ( x ) = x \Longleftrightarrow x ^ { 2 } - 5 x + 4 = 0 .$$

b. Solve $f ( x ) = x$ in the interval $] - \infty$; 5[.\\
  \item Prove that the sequence ( $u _ { n }$ ) is convergent.\\
Determine its limit.\\
  \item Would the behavior of the sequence be identical by choosing as initial term $u _ { 0 } = 4$ instead of $u _ { 0 } = 3$ ?
\end{enumerate}