A question about the expansion in base $g$ of real numbers

Question

I'm reading the proof of Theorem 7.11 from textbook Analysis I by Amann/Escher.

For $x \in [0,1)$ the authors define recursively the sequence $(x_k)_{k \in \mathbb N}$ as follows:

In the final part, the authors prove that if $x \in [0,1)$ is rational then the base $g$ expansion of $x$ is periodic.

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I have two question regarding this last part:

1. Is $r_{j_{0}+1}=r_{k_{0}}$ a typo? I think it should be $r_{j_{0}}=r_{k_{0}}$.

2. How does it follow that $r_{j_{0}+i}=r_{k_{0}+i}$ for all $1 \leq i \leq j_{0}-k_{0}$? I tried to use $r_{k} :=g r_{k-1}-x_{k}$ but stuck since $x_k$ is in the formula.

Thank you for your help!

Answer 1

1. Yes it is a typo. The correct one is $r_{j_0+1}=r_{k_0+1}$.

2. It follows from $r_{k+1}=gr_k-x_{k+1}=gr_k-\lfloor gr_k\rfloor$ for all $k$. Hence, if $r_i=r_j$ then $r_{i+1}=r_{j+1}$.