Here is Theorem 12.1 of Introduction to Analytic Number Theory by Apostol -
Theorem 12.1 The series for $ \zeta(s, a) $ converges absolutely for $ \sigma > 1 $. The convergence is uniform in every half-plane $ \sigma \ge 1 + \delta $, $ \delta > 0 $, so $ \zeta(s, a) $ is an analytic function of $ s $ in the half-plane $ \sigma > 1 $.
Proof. All these statements follow from the inequalities
$$ \sum_{n=1}^{\infty} |(n + a)^{-s}| = \sum_{n=1}^{\infty} (n + a)^{-\sigma} \le \sum_{n=1}^{\infty} (n + a)^{-(1 + \delta)}. \tag*{$\Box$}$$
How does this proof work? The last sum on the right is
$$ \frac{1}{(1 + a)^{\delta + 1}} + \frac{1}{(2 + a)^{\delta + 1}} + \frac{1}{(3 + a)^{\delta + 1}} + \dots. $$
where $ \delta > 0 $ and $ 0 < a < 1 $. Why does this converge? I can't find anything in the preceding sections or chapters that explain why this must converge. How can we show that this sum converges?