$\newcommand{\Z}{\mathbf Z}\newcommand{\Q}{\mathbf Q}$I am currently reading Serre's A Course in Arithmetic, and in Chapter 2 where he introduces the $p$-adics, he mentions the projective limit. My question is not so much about the projective limit itself, but about the notation he uses.
First, for context, he constructs the $p$-adic integers $\Z_p$ as a certain projective limit, then the full $p$-adic numbers $\Q_p$ as the field of fractions of $\Z_p$. Here is the relevant excerpt from the text.
So far so good; as you can see, he uses $\varprojlim(A_n,\phi_n)$ for the projective limit, which is in agreement with what I've been able to find elsewhere and seems to be common notation. However, a few pages later, he starts to use some notation that I do not recognise and is seemingly inconsistent.
For example, he writes $\Z_p=\varprojlim\!\mathbf{.} A_i$ where you can see the clear "dot" between the limit symbol and $A_i$. Is this supposed to mean that the maps $\phi_i$ are omitted from notation, or something else? If so, why the dot symbol?
Similarly, for a sequence of $(u_n)\in\Q_p$, he says that "a sequence $u_n$ has a limit if and only if $\lim\!\mathbf{.}(u_{n+1}-u_n)=0$". Again there is the appearance of this conspicuous dot. For the rest of the text, as far as I see, this dot persists in all notation in this context.
Question. What's going on with this dot? Does $\varprojlim\!\mathbf{.}$ mean the same as $\varprojlim$, and does $\lim\!\mathbf{.}$ mean the same as $\lim$?
I am probably missing something obvious here; many thanks in advance!