We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. 15a. Show that the sequence $\left( F _ { i } \right) _ { i \geqslant 0 }$ converges for the norm $\| \cdot \| _ { r , s }$ towards a monic polynomial $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ of degree $d$ which satisfies $F _ { \mid t = 0 } = X ^ { d }$. 15b. Show that there exists $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ such that $P = F G$.
We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula:
$$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$
where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$.
\textbf{15a.} Show that the sequence $\left( F _ { i } \right) _ { i \geqslant 0 }$ converges for the norm $\| \cdot \| _ { r , s }$ towards a monic polynomial $F \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ of degree $d$ which satisfies $F _ { \mid t = 0 } = X ^ { d }$.
\textbf{15b.} Show that there exists $G \in \mathscr { D } _ { r } \left( \mathbb { R } _ { n } [ X ] \right)$ such that $P = F G$.