Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r < \rho$. Let $\left( f _ { n } \right) _ { n \geqslant 0 }$ be a sequence of elements of $\mathscr { D } _ { \rho } ( \mathbb { R } )$. We assume that $\sum _ { n \geqslant 0 } \left\| f _ { n } \right\| _ { r }$ converges. Show that $\sum _ { n \geqslant 0 } f _ { n }$ converges normally on $U _ { r }$ to a function $f \in \mathscr { D } _ { r } ( \mathbb { R } )$. Show that $\sum _ { n \geqslant 0 } f _ { n }$ also converges to $f$ for the norm $\| \cdot \| _ { r }$.
Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r < \rho$. Let $\left( f _ { n } \right) _ { n \geqslant 0 }$ be a sequence of elements of $\mathscr { D } _ { \rho } ( \mathbb { R } )$. We assume that $\sum _ { n \geqslant 0 } \left\| f _ { n } \right\| _ { r }$ converges. Show that $\sum _ { n \geqslant 0 } f _ { n }$ converges normally on $U _ { r }$ to a function $f \in \mathscr { D } _ { r } ( \mathbb { R } )$. Show that $\sum _ { n \geqslant 0 } f _ { n }$ also converges to $f$ for the norm $\| \cdot \| _ { r }$.