%
\label{thm:real-inverse-limit}
Let $\ell_\alpha^{\mathrm{den}}
:= \bigl\{\mathrm{den}(\alpha_n) : n \geq 1\bigr\}$
denote the image of the $\alpha$-ray
under the denotation map
(Definition~\textup{\ref{def:ch22-denotation-map}},
Chapter~\textup{\ref{ch:orthodox-bridge}}).
Then:
\begin{enumerate}
    \item[\textup{(i)}]
          $\ell_\alpha^{\mathrm{den}}$
          is a countable dense subset
          of a closed interval $[a, b] \subset \mathbb{R}$.
    \item[\textup{(ii)}]
          The closure
          $\overline{\ell_\alpha^{\mathrm{den}}}$
          in the Euclidean topology
          is isometric to $[a, b]$,
          hence homeomorphic to~$[0, 1]$.
    \item[\textup{(iii)}]
          $\mathbb{R}$ is recovered as the
          \textbf{Archimedean inverse limit}
          \[
              \boxed{%
              \mathbb{R}
              \;\cong\;
              \varprojlim_{k}\;
              \frac{1}{P_k}\,\mathbb{Z}
              \;=\;
              \Bigl\{\,
              (r_k)_{k \geq 1}
              \;\Big|\;
              r_k \in \tfrac{1}{P_k}\mathbb{Z},\;
              r_k \equiv r_{k+1} \!\!\pmod{1/P_k}
              \,\Bigr\}.}
          \]
          Every element of this inverse limit
          corresponds to a unique real number:
          the common value of the nested intervals
          $\bigl[r_k, r_k + 1/P_k\bigr)$.
\end{enumerate}

\textbf{(i).}
The image points
$\mathrm{den}(\alpha_n) = D(\alpha_n)/P_n$
are rational numbers in~$[0, 1)$
(or a shifted interval, depending on the normalization convention).
For distinct stages $m < n$,
the CRT coherence guarantees that
$D(\alpha_n)/P_n$ lies within $1/P_m$
of $D(\alpha_m)/P_m$.
The points accumulate within a single interval
$[a, b]$ of length at most~$1$.
Density follows because for any rational $q/P_k$
in the interval,
some deeper-stage point $\alpha_n$ with $n \geq k$
has $D(\alpha_n)/P_n$ within $1/P_k$ of $q/P_k$.

\smallskip
\noindent\textbf{(ii).}
The closure of a countable dense subset
of a closed interval
in the Euclidean topology
is the interval itself.
The completeness of~$\mathbb{R}$
fills the gaps between the rational grid points.

\smallskip
\noindent\textbf{(iii).}
Define the inverse system:
the $k$th object is the grid
$\frac{1}{P_k}\mathbb{Z} \cap [a, b]$,
and the transition map
$\varphi_{k, k+1} :
\frac{1}{P_{k+1}}\mathbb{Z} \to
\frac{1}{P_k}\mathbb{Z}$
sends $r_{k+1}$ to the nearest grid point
$r_k$ with $r_k \leq r_{k+1}$.
This is well-defined because $P_k \mid P_{k+1}$.
A coherent sequence $(r_k)_{k \geq 1}$
determines a nested sequence of intervals
$[r_k, r_k + 1/P_k)$,
whose intersection is a single real number
(by Archimedean completeness:
$1/P_k \to 0$).
Conversely, every real number $x \in [a, b]$
determines a unique coherent sequence
by setting $r_k = \lfloor x \cdot P_k \rfloor / P_k$.
The bijection is order-preserving
and respects the Euclidean metric.