The purpose of Part A is to study the behavior of the sequence $\left( u _ { n } \right)$ defined by $u _ { 0 } = 0.3$ and by the recurrence relation, for all natural integer $n$ :

$$u _ { n + 1 } = 2 u _ { n } \left( 1 - u _ { n } \right)$$

This recurrence relation is written $u _ { n + 1 } = f \left( u _ { n } \right)$, where $f$ is the function defined on $\mathbb { R }$ by :

$$f ( x ) = 2 x ( 1 - x )$$

\begin{enumerate}
  \item Prove that the function $f$ is strictly increasing on the interval $\left[ 0 ; \frac { 1 } { 2 } \right]$.
  \item We admit that for all natural integer $n , 0 \leqslant u _ { n } \leqslant \frac { 1 } { 2 }$.\\
Calculate $u _ { 1 }$ then perform a proof by induction to demonstrate that for all natural integer $n , u _ { n } \leqslant u _ { n + 1 }$.
  \item Deduce that the sequence $( u _ { n } )$ is convergent.
  \item Justify that the limit of the sequence $( u _ { n } )$ is equal to $\frac { 1 } { 2 }$.
\end{enumerate}