I am reading the proof of the following proposition from Stein and Shakarchi's Complex Analysis. The proof below applies the mean-value theorem to the complex-valued function $f(x)=x^{-s}$, where $s \in \mathbb{C}$. How is this justified? I could not find any source mentioning the mean-value theorem for functions from the real domain to the complex plane.
Next, the book uses this proposition to prove the following corollary. In the proof, I don't understand how we have the uniform convergence of the series $\sum \delta_n(s)$ in any half plane $Re(s) \ge \delta, \delta >0$, by the estimate $|\delta_n(s)| \le |s|/n^{\sigma +1}$. Perhaps they mean uniform convergence over compact sets of the half-space? Because this is all we need to extend holomorphic of an infinite sum, so I think this suffices. But I don't see how we can bound $|s|$ on the entire half-space to use Weierstrass' M-test, or any other criteria for uniform convergence of a series. I would greatly appreciate any explanation on these questions.
The mean-value theorem, in the context of Complex Analyisis, states that if $f\colon[a,b]\longrightarrow\mathbb C$ is differentiable, then, for some $c\in[a,b]$,$$\left\lvert\frac{f(b)-f(a)}{b-a}\right\rvert\leqslant\bigl\lvert f'(c)\bigr\rvert.$$That is exactly what is being used here.