# Real numbers as n-ary fractions

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.

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The cut symbol is a typo. This is supposed to be $m_k n \le m_{k+1} \le m_k n + n - 1$.