Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) \neq 0$. The purpose of this question is to show that there exists $r \in \mathbb { R } _ { + } ^ { * } , r \leqslant \rho$ such that $\frac { 1 } { f } \in \mathscr { D } _ { r } ( \mathbb { R } )$. 6a. Show that we can assume without loss of generality that $f ( 0 ) = 1$. We now write $f ( t ) = \sum _ { i = 0 } ^ { \infty } a _ { i } t ^ { i }$ and we assume that $a _ { 0 } = 1$. 6b. Only in this sub-question, we assume that there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $g \in \mathscr { D } _ { r } ( \mathbb { R } )$ such that $f ( t ) g ( t ) = 1$ for all $t \in U _ { r }$. We write $g ( t ) = \sum _ { i = 0 } ^ { \infty } b _ { i } t ^ { i }$. Show that: $$\left\{ \begin{aligned} & b _ { 0 } = 1 \\ & \text { for } n \geqslant 1 , b _ { n } = - \left( b _ { 0 } a _ { n } + \ldots + b _ { n - 1 } a _ { 1 } \right) \end{aligned} \right.$$ We now define the sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ by the above recurrence formula. 6c. Show that there exists $c \in \mathbb { R } _ { + } ^ { * }$ such that $\left| a _ { n } \right| \leqslant c ^ { n }$ for all $n \in \mathbb { N }$. 6d. Show that $\left| b _ { n } \right| \leqslant ( 2 c ) ^ { n }$ for all $n \in \mathbb { N }$. 6e. Conclude.
Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) \neq 0$. The purpose of this question is to show that there exists $r \in \mathbb { R } _ { + } ^ { * } , r \leqslant \rho$ such that $\frac { 1 } { f } \in \mathscr { D } _ { r } ( \mathbb { R } )$.
\textbf{6a.} Show that we can assume without loss of generality that $f ( 0 ) = 1$.
We now write $f ( t ) = \sum _ { i = 0 } ^ { \infty } a _ { i } t ^ { i }$ and we assume that $a _ { 0 } = 1$.
\textbf{6b.} Only in this sub-question, we assume that there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $g \in \mathscr { D } _ { r } ( \mathbb { R } )$ such that $f ( t ) g ( t ) = 1$ for all $t \in U _ { r }$.
We write $g ( t ) = \sum _ { i = 0 } ^ { \infty } b _ { i } t ^ { i }$. Show that:
$$\left\{ \begin{aligned} & b _ { 0 } = 1 \\ & \text { for } n \geqslant 1 , b _ { n } = - \left( b _ { 0 } a _ { n } + \ldots + b _ { n - 1 } a _ { 1 } \right) \end{aligned} \right.$$
We now define the sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ by the above recurrence formula.
\textbf{6c.} Show that there exists $c \in \mathbb { R } _ { + } ^ { * }$ such that $\left| a _ { n } \right| \leqslant c ^ { n }$ for all $n \in \mathbb { N }$.
\textbf{6d.} Show that $\left| b _ { n } \right| \leqslant ( 2 c ) ^ { n }$ for all $n \in \mathbb { N }$.
\textbf{6e.} Conclude.