For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
- Consider the function $f$ defined on $\mathbb { R }$ by: $f ( x ) = 5 x \mathrm { e } ^ { - x }$.
We denote by $C _ { f }$ the representative curve of $f$ in an orthonormal coordinate system.
Statement 1: The $x$-axis is a horizontal asymptote to the curve $C _ { f }$.
Statement 2: The function $f$ is a solution on $\mathbb { R }$ of the differential equation $( E ) : y ^ { \prime } + y = 5 \mathrm { e } ^ { - x }$.
2. Consider the sequences $\left( u _ { n } \right) , \left( v _ { n } \right)$ and $\left( w _ { n } \right)$, such that, for every natural integer $n$ :
$$u _ { n } \leqslant v _ { n } \leqslant w _ { n } .$$
Moreover, the sequence $( u _ { n } )$ converges to $- 1$ and the sequence $( w _ { n } )$ converges to $1$.
Statement 3: The sequence $\left( \nu _ { n } \right)$ converges to a real number $\ell$ belonging to the interval $[ - 1 ; 1 ]$.
We further assume that the sequence $( u _ { n } )$ is increasing and that the sequence $( w _ { n } )$ is decreasing.
Statement 4: For every natural integer $n$, we then have: $\quad u _ { 0 } \leqslant v _ { n } \leqslant w _ { 0 }$.