I am following a proof from a book, but it seems to have a typo or use some strange notation I have never seen, and I have already spent a lot of time trying to decipher this, but cannot find the solution.
It is basically showing one how to construct general real numbers from Dedekind's definition by expressing those as a finite or infinite n-ary function.
So, it is trying to prove that any real number $x = a_{0}\cdot a_{1}a_{2}a_{3}.... = a_{0}+\frac{a_{1}}{n} + \frac{a_{2}}{n^2} + ...$ with $ 0\leq a_{k}\leq n-1 $ for $k=1,2,3,...$.
It starts by pointing out that $x$ being irrational, it can be defined as the Dedekind cut $x=D_{a}|D_{b}$. Then it considers the sets
$C_{k}=\left \{ \frac{m}{n^k} \right \}$ , $m=...,-2,-1,0,1,2...$
such that $C_{0}\subset C_{1}\subset C_{2} \subset ... \subset C_{k} \subset C_{k+1}$. In each of these there is a largest number $m_{k}$ such that $m_{k}/n^k$ belongs to $D_{a}$ while $m_{k}+1/n^k$ belongs to $D_{b}$ so that
$ \frac{m_{k}}{n^k}\leq x \leq \frac{m_{k}+1}{n^k} $
After this it says that since $C_{k} \subset C_{k+1}$, it follows that
$m_{k}|n \leq m_{k+1} \leq m_{k} n+(n+1)$
and in the author's words "that is, $m_{k+1}=m_{k}n+a_{k+1}$, where $0 \leq a_{k+1}\leq n -1$".
So it says that this completes the informal proof, since by choosing k=0,1,2,..., this leads to an algorithm generating n-ary fraction.
I get lost after the part that says "since $C_{k} \subset C_{k+1}$", as in the equation that follows this the author uses the cut symbol |, for $m_{k}$, and at the last inequality there is no mathematical operator between $m_{k}$ and n + (n+1) and I assume a product is not involved since there are no parenthesis. Same thing happens in the equation below between and $m_{k}$ and $n$, where in the book there is even a visible space between the two numbers.
Any hints on what happens from the point I got lost on will be extremely helpful.
The book I am reading is "Linear Algebra for Quantum Theory" by Per-Olov Lowdin.