Dirichlet Series Absolute Convergence Lemma

Theorem
Let $\displaystyle f(s)= \sum_{n=1}^\infty \frac{a_n}{n^s}$ be a Dirichlet series.

Suppose that $f$ converges absolutely at $s_0 = \sigma_0 + i t_0 \in \C$.

Then $f$ converges absolutely at all points $s = \sigma + i t \in \C$ with $\sigma \geq \sigma_0$.

Proof
Suppose that $f$ converges absolutely at $\sigma_0 + i t_0$.

If $\sigma \geq \sigma_0$, then

Therefore absolute convergence of $f(s_0)$ directly implies absolute convergence of $f(s)$ for all $s=\sigma + it$ with $\sigma > \sigma_0$.