Dealing with the $p$-adic numbers with their canonic metric $(\mathbb{Q}_p,\|\|_{p})$, one usually defines $$(\mathbb{Q}_\infty,\|\|_{\infty}):= (\mathbb{R},\|\|). \tag{1}$$
While in the first instance this can be seen as barely a definition, there is ample evidence that it is a very sensible, if not the only proper way to define things. Maybe the most notable evidence is
$$ \prod_{p\le\infty} \|\alpha\|_p = 1 \tag{2}$$ for any rational $\alpha$.
Now this suggests that there might be a way to carry out a limit operation such that (2) is the evaluation of a
$$ \lim_{q\rightarrow\infty} \prod_{p=2}^{q}\|\alpha\|_p = \prod_{p\le\infty} \|\alpha\|_p$$
and instead of $(1)$ it is much more $$\lim_{p\rightarrow\infty} (\mathbb{Q}_p,\|\|_{p}) = (\mathbb{R},\|\|),$$ which puzzles me a lot.
(Moreover one notes that the commonly found $(2)$ is not exactly very "clean" as well in dealing with a improper inequality involving the $\infty$ symbol.)
For example
- when looking at the definition of the $p$-adic metric of a number $x=\frac{a}{b}p^{\nu_p(x)}$ (with the suitably chosen $a,b,p$ and $x\ne0$) $$ \|x\|_p:=p^{-\nu_p(x)}.$$ Is there any way to see how such a limit $p\rightarrow\infty$ could work here to get the normal Euklidian metric?
- Another puzzle for me in this is that all the $p$-adic metrics are obviously discrete, while the Euclidian metric of is continuous. So how can this work when we have this $\lim_{p\rightarrow\infty}$?
It appears to me that this is related to an inverse limit construction of the reals from the $p$-adics, a topic which is discussed for example here.
In summary the question is, if there is a limit construction possible, e.g. employing an inverse or a direct limit, for constructing $\mathbb{R}$ from the set of $\mathbb{Q}_p$?