About the uniqueness of decimal expression of a real number 
Definition


A sequence $(q_n)_{n\in\Bbb N}$ of rational numbers is said a Cauchy sequence if for any $\epsilon\in\Bbb Q^+$ there exist $n_0\in\Bbb N$ such that
$$
|q_m-q_n|<\epsilon
$$
for any $m,n\ge n_0$. In particular two Cauchy sequences $(a_n)_{n\in\Bbb N}$ and $(b_n)_{n\in\Bbb N}$ are said $c$-equivalent if for any $\epsilon\in\Bbb Q^+$ there exists $n_0\in\Bbb N$ such that
$$
|a_n-b_n|<\epsilon
$$
for any $n\ge n_0$

So it is not hard the above relation $\underset{c}\sim$ of $c$-equivalence is an equivalence relation so that it is possible to put
$$
\Bbb R:=\mathcal C/\underset{c}\sim
$$
For details about the last definition (of real numbers set) we refer to the text Set Theory and Logic by Robert R. Stoll.
So I'd like to discuss the proof of the following theorem which is taken from the mentioned text.

Theorem
Let $r$ be an integer greater than or equal to 2. Corresponding to each nonnegative real number $x$ there is a sequence $(d_n)_{n\in\Bbb N}$ of integers which is uniquely determined by $x$ (relative to $r$) such that

*

*$d_0=\lfloor x\rfloor$ the largest integer less than or equal to $x$

*$d_n\in[0,r)$ for any $n\in\Bbb N$

*the sequence whose terms are defined inductively by
$$
y_0=d_0\\
y_{n+1}=y_n+\frac{d_{n+1}}{r^{n+1}}
$$
is a Cauchy sequence whose limit is $x$
Proof. Let $r$ be an integer greater that or equal to $2$, $x$ be a nonnegative real number and $d_0=\lfloor x\rfloor$. Then
$$
xr=ar+x_1
$$
for some number $x_1\in[0,r)$. Let $d_1:=\lfloor x_1\rfloor$ so
$$
x_1r=d_1r+x_2
$$
for some number $x_2$ such that $x_2\in[0,r)$. Let $d_2:=\lfloor x\rfloor$ so
$$
x_2r=d_2r+x_3
$$
for some number $x_3\in[0,r)$. In general, define $x_n$ by
$$
x_{n-1}r=d_{n-1}r+x_n
$$
and set $d_n=\lfloor x\rfloor$. Then
$$
x=d_0+\frac{d_1}{r}+\frac{d_2}{r^2}+\dots+\frac{d_n}{r^n}+\frac{x_{n+1}}{r^{n+1}}
$$
where $x_{n+1}\in[0,r)$. Hence
$$
0\le x-\Biggl(d_0+\frac{d_1}{r}+\frac{d_2}{r^2}+\dots+\frac{d_n}{r^n}\Biggl)<\frac{1}{r^n}
$$
According to the definition of $y_n$ given in $(3)$ this may be written as
$$
0\le x-y_n<\frac{1}{r^n}
$$
It follows that $|x-y_n|<\frac{1}{r^n}$ whence $\lim_{n\in\Bbb N} y_n=x$.
The proof of the uniqueness (relative to $r$) of the sequence corresponding to $x$ is left as an exercise

Moreover even the converse of the preceding theorem is true, that is

Let $(d_n)_{n\in\Bbb N}$ be a sequence of nonnegative integers such that for some integer $r\ge 2$ and $d_n\in[0,r)$ for all natural numbers $n$. Then there exists a unique nonnegative real number $x$ such that the sequence whose terms $y_n$ are defined inductively by $y_0=d_0$ and $y_{n+1}=y_n+\frac{d_{n+1}}{r^{n+1}}$ is a Cauchy sequence having $x$ as its limit.

So unfortunately I did not understand what is the sequence corresponding to $x$ relative to $r$ so that I did not able to prove its uniqueness too. Anyway I think that this sequece cannot be the sequece $(d_n)_{n\in\Bbb N}$ because when $r=10$ it is well know that
$$
0.999999999\dots=1=1.000000000\dots
$$
So could someone explain to me which is the sequence corresponding to $x$ relative to $r$ and why this sequence is unique? Perhaps is it possible that the sequence $(d_n)_{n\in\Bbb N}$ is unique if the terms greater that $0$ are not all equal? Could someone help me, please?
For sake of completness here the original text of the last theorem.
 A: The theorem is technically not wrong: it simply does not (in my opinion) state that decimal expansion (or expansion in any base) is unique.
The statement is "there is a sequence $(d_n)n∈\mathbb{N}$ of integers which is uniquely determined by $x$ [...]". It does not say that it is the only possible sequence. Just that we can define a sequence $(d_n)$ in a non-ambiguous way, knowing only $x$. Here the construction chooses to take the sequence ending with an infinity of $0$s instead of an infinity of $9$s in the case of a decimal number (or the equivalent in base $r$). This sequence is uniquely determined by $x$ and it satisfies all the properties, as the theorem states.
This being said, I do agree that the formulation is ambiguous, and maybe I'm trying too hard to salvage it. In particular, it is a crucial ommission from the author to not include a comment on the fact that decimal numbers admit two expansions. But if I had to be a judge on this statement's trial, I would have to rule that it is technically not incorrect.
