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 $n \in \mathbb{N}$: $$B _ { n } = \sum _ { k = 0 } ^ { n } b _ { k } .$$ We assume in this question that there exists a sequence $\left( m _ { n } \right) _ { n \in \mathbb{N} }$ satisfying $m _ { n } > n$ for $n$ large enough and $$B _ { m _ { n } - n } \underset { n \rightarrow + \infty } { \sim } B _ { n } \quad \text{and} \quad B _ { m _ { n } } - B _ { m _ { n } - n } \underset { n \rightarrow + \infty } { \longrightarrow } 0 .$$ Show that $$a _ { n } \underset { n \rightarrow + \infty } { \sim } \frac { 1 } { B _ { n } } .$$