Dirichlet's Test Remark in Apostol Dirichlet's Test is theorem $10.17$ in Apostol's Calculus Vol. $1$.
The theorem itself says that if the partial sums of $\{a_n\}$ (can be complex numbers, not just reals) form a bounded sequence and $\{b_n\}$ is a (monotone?) decreasing function converging to $0$, then $\sum a_n b_n$ converges.
The part of the proof I am stuck on says that, letting $A_n=\sum_{k=1}^{n} a_k$
"The series $\sum (b_k - b_{k+1})$ is a convergent telescoping series which dominates $\sum A_k(b_k - b_{k+1})$.  This implies absolute convergence..."
How does this imply absolute convergence?  Does it have to do with the fact that $\{b_n\}$ is decreasing?  By decreasing, should I automatically think monotone?

The could be a typo in the book, where the absolute value is missing, because $A_k$ could be complex, so the telescopic series actually dominates $\sum |A_k (b_k - b_{k+1})|$. Otherwise, I don't see how the relationship even makes sense, because $\le$ is not defined between complex numbers. The rest of the proof is the same as in the book.
