Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a real decreasing sequence converging to 0, and $\left( b _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a complex sequence such that the sequence $\left( B _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ defined for all $n \in \mathbb { N } ^ { * }$ by $B _ { n } = b _ { 1 } + \cdots + b _ { n }$ is bounded. Show that, for all integer $n \geqslant 2$, $$\sum _ { k = 1 } ^ { n } a _ { k } b _ { k } = a _ { n } B _ { n } + \sum _ { k = 1 } ^ { n - 1 } \left( a _ { k } - a _ { k + 1 } \right) B _ { k }$$
Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a real decreasing sequence converging to 0, and $\left( b _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be a complex sequence such that the sequence $\left( B _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ defined for all $n \in \mathbb { N } ^ { * }$ by $B _ { n } = b _ { 1 } + \cdots + b _ { n }$ is bounded.
Show that, for all integer $n \geqslant 2$,
$$\sum _ { k = 1 } ^ { n } a _ { k } b _ { k } = a _ { n } B _ { n } + \sum _ { k = 1 } ^ { n - 1 } \left( a _ { k } - a _ { k + 1 } \right) B _ { k }$$