For $p \in \Bbb Z $ there is a q-ary system described which assigns to each real number x in the base q a sequence of {$\alpha_n$} such that $\sum_{i=0}^n \alpha_{p-i}\cdot q^{p-i} \le x \lt q^{p-n} + \sum_{i=0}^n \alpha_{p-i}\cdot q^{p-i}$ where $\alpha_{p-i} \in \{0,1,\ldots,q-1\}$ and thus $$ r_n = \alpha_pq^p + \ldots + \alpha_{p-n}q^{p-n} $$ but later on he states and proves that (page 63-64):
We remark that by virtue of the algorithm just described for obtaining the numbers $\alpha_{p - n} \in \{ 0 , 1 , . . . , q — 1\}$ successively, it cannot happen that all these numbers from some point on are equal to q — 1.
With the proof being:
if $$ r_n = \alpha_pq^p + \ldots + \alpha_{p-k}q^{p-k}+(q-1)q^{p-k-1}+ \ldots +(q-1)q^{p-n} $$ for all n > k, that is, $$r_n = r_k + \frac1{q^{k-p}} - \frac1{q^{n-p}}$$ But from the definition of the system $$\sum_{i=0}^n \alpha_{p-i}\cdot q^{p-i} \le x \lt q^{p-n} + \sum_{i=0}^n \alpha_{p-i}\cdot q^{p-i}$$ thus $$ r_n \le x < r_n + q^{p-n}$$ $$\implies r_k + \frac1{q^{k-p}} - \frac1{q^{n-p}} \le x < r_k + \frac1{q^{k-p}}$$ Then for any n > k $$ 0 < r_k + \frac1{q^{k-p}} -x < \frac1{q^{n-p}}$$ Which is impossible because if the number $h \in R$ is such that 0 $\le$ h and $h < \frac1n$ for all $n \in \Bbb N$, then h = 0. Here h is strictly greater than 0.
But if this is the case then what about the number 9999999 in base 10, which will make all $\alpha$ to be q-1 = 10-1 = 9, or any number that has 9s after a certain position. How will such numbers be represented in the system?
It's talking about infinite ending in decimal fractions or its counterpart in number system of base $q$.
Note that $p$ (or rather, $p+1$) is the number of digits until the decimal point, as the exponent in $q^{p-n}$ becomes negative as $n>p$.
Consider for example $1.23999999\overset\cdot9$ which equals $1.24$.
Here we have $q=10,\ p=1$ and $\alpha_2$ can't be $3$ because of the given constraint, must be $4$.
This is done exactly in order to eliminate the infinite ending of the digit $q-1$, and thus obtain a pure bijection to the real numbers.