Let $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ and $\left( b _ { n } \right) _ { n \in \mathbb{N} }$ be two sequences of elements of $\mathbb{R}^{+*}$. We assume that $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ is decreasing and that $$\forall n \in \mathbb{N} , \quad \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } = 1 .$$ We set, for $n \in \mathbb{N}$: $$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$ Let $m$ and $n$ be two natural integers such that $m > n$. Show that $$a _ { n } \leq \frac { 1 } { B _ { n } } \quad \text{and} \quad 1 \leq a _ { n } B _ { m - n } + a _ { 0 } \left( B _ { m } - B _ { m - n } \right) .$$
Let $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ and $\left( b _ { n } \right) _ { n \in \mathbb{N} }$ be two sequences of elements of $\mathbb{R}^{+*}$. We assume that $\left( a _ { n } \right) _ { n \in \mathbb{N} }$ is decreasing and that
$$\forall n \in \mathbb{N} , \quad \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } = 1 .$$
We set, for $n \in \mathbb{N}$:
$$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$
Let $m$ and $n$ be two natural integers such that $m > n$. Show that
$$a _ { n } \leq \frac { 1 } { B _ { n } } \quad \text{and} \quad 1 \leq a _ { n } B _ { m - n } + a _ { 0 } \left( B _ { m } - B _ { m - n } \right) .$$