Assume the following is true
$$\left|\sum_{x_{1}<n\leq x_{2}}\frac{a_{n}}{n^{s}}\right| \leq Kx_{1}^{-\sigma}$$
where $s=\sigma+it$ and $a_n$ are complex, and all other variables are real, and $\sigma>0$.
We say that the magnitude of the sum on the LHS is bounded. Being bounded doesn't necessarily mean it converges.
If $x_1 \rightarrow \infty$ then the RHS tends to zero. The bound on the LHS tends to zero.
Question: Why does this mean the sum $\sum_{x_{1}<n\leq x_{2}}a_{n}n^{-s}$ converges?
Notes: If the the magnitude of individual terms went to zero, this would not be considered a sufficient (although it is necessary) for convergence. Why is it sufficient that the magnitude of the tail $\sum_{x<n\leq\infty}a_{n}n^{-s}$ of the series tends to zero?
This question is about convergence and bounds. The context is Apostol's proof that a Dirichlet Series has an abscissa of convergence. Apostol IANT section 11.6 pages 232-233. I think the question is independent of this context, but is referenced in case I am wrong.