The Bloch space $\mathcal{B}$ consist of all analytic fuctions $f$ defined on the open unit disc $\mathbb{D}$ such that \begin{align} ||f||_\mathcal{B}:=\sup_{|z|<1}(1-|z|^2)|f'(z)|<\infty. \end{align} The Bloch norm is defined as $||f|| = ||f||_\mathcal{B} + |f(0)|$.
According to the book Bergman Spaces by P. Duren and A. Schuster (proposition 1, chapter 2.6) the following inequality is satisfied: \begin{align} |f(z)| \leq ||f|| \log \frac{1+|z|}{1-|z|}, \; \; \forall\; |z|\geq \frac{1}{2}. \end{align} Then, when they prove that $\mathcal{B}$ is complete, they say that if one has a Cauchy sequence $\{f_n\}$ in $\mathcal{B}$, then by the previous inequality it is a uniform Cauchy sequence on each compact subset of $\mathbb{D}$ and thus it is uniformly convergent in $\mathbb{D}$.
I do not understand how this inequality implies that the sequence is a uniform Cauchy sequence on each compact subset of $\mathbb{D}$.
The reason they give is:
The inequality implies that the pointwise evaluation is a bounded linear functional on the Bloch space, with a norm that is uniformly bounded on each compact subset of $\mathbb{D}$. This, together with the maximum principle, implies that if a sequence of functions converges in the Bloch norm, then it does so locally uniformly.
But I do not follow all these conclusions. Could you explain me these consequences? I would appreciate it very much because I really want to understand this proof.