I am self-learning Real Analysis from the text, Understanding Analysis, by Stephen Abbott. I would like to prove Abel's test of convergence in the following exercise problem 2.7.12. A clue/hint, without revealing the entire solution would be extremely helpful.
I reproduce my attempt for part(a) of the problem below.
(Abel's Test). Abel's test for convergence states that if the series $\displaystyle \sum _{k=1}^{\infty } x_{k}$ converges and if $\displaystyle ( y_{k})$ is a sequence satisfying:
\begin{equation*} y_{1} \geq y_{2} \geq y_{3} \geq \dotsc \geq 0 \end{equation*} then the series $\displaystyle \sum _{k=1}^{\infty } x_{k} y_{k}$ converges.
(a) Use exercise 2.7.12 to show that:
\begin{equation*} \sum _{k=1}^{n} x_{k} y_{k} =s_{n} y_{n+1} +\sum _{k=1}^{n} s_{k}( y_{k} -y_{k+1}) \end{equation*}
where $\displaystyle s_{n} =x_{1} +\dotsc +x_{n}$.
Proof.
By the formula for summation by parts, we have:
\begin{equation*} \begin{array}{ c l } \sum _{k=1}^{n} x_{k} y_{k} & =s_{n} y_{n+1} -s_{0} y_{1} +\sum _{k=1}^{n} s_{k}( y_{k} -y_{k+1})\\ & =s_{n} y_{n+1} -0\cdot y_{1} +\sum _{k=1}^{n} s_{k}( y_{k} -y_{k+1})\\ & =s_{n} y_{n+1} +\sum _{k=1}^{n} s_{k}( y_{k} -y_{k+1}) \end{array} \end{equation*}
(b) Use the comparison test to argue that: $\displaystyle \sum _{k=1}^{\infty } s_{k}( y_{k} -y_{k+1})$ converges absolutely, and show how this leads directly to a proof of Abel's test.