Sum of Sequence as Summation of Difference of Adjacent Terms/Proof 2

Proof
From Abel's Lemma, after renaming and reassigning variables:


 * $\displaystyle \sum_{k \mathop = 1}^n a_k b_k = \sum_{k \mathop = 1}^{n - 1} A_k \left({a_k - a_{k + 1} }\right) + A_n a_n$

where:
 * $\left \langle {a} \right \rangle$ and $\left \langle {b} \right \rangle$ be sequences in $\R$
 * $\displaystyle A_n = \sum_{k \mathop = 1}^n {b_k}$ be the partial sum of $\left \langle {b} \right \rangle$ from $1$ to $n$.

Let $\left \langle{b}\right \rangle$ be defined as:
 * $\forall k: b_k = 1$

Thus:
 * $\displaystyle A_n = \sum_{k \mathop = 1}^n 1 = n$

and so: