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?