In Complex Analysis by Stein and Shakarchi, on page 173 the authors state that
$\left| \delta_n( s) \right|\leq |s|/n^{\sigma+1}$
implies that the series $\sum\delta_n(s)$ converges uniformly on any half-plane $Re(s)\geq\delta>0$ (here $\sigma:=Re(s)$, and $\delta_n(s)$ is defined to be the integral $\int_n^{n+1}\left[\frac{1}{n^s}-\frac{1}{x^s} \right]dx$).
I am not sure how the authors are getting uniform convergence from the above bound. I know we can use the Weierstrass M-Test to obtain uniform convergence, but that doesn't seem to be applicable here since $|s|/n^{\sigma+1}$ is dependent on $s$. In fact, it seems as though $|s|/n^{\sigma+1}$ can be made arbitrarily large by choosing $Im(s)$ to be arbitrarily large.
How do we know that the series converges uniformly? Any help is appreciated.