TITLE: Going from completion to explicit description of the real numbers
QUESTION [7 upvotes]: I have understood, I think, the construction of the real numbers as the set of equivalence classes of Cauchy sequences. That is, if $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences, then we say they are equivalent if $\lim_{n\to \infty} \lvert a_n - b_n\rvert = 0$. The set of equivalence classes is then a field under the "obvious" operations. This is what I believe is the completion of $\mathbb{Q}$ with respect to the absolute value.
How does one go from this to the fact that any real number can be described as "infinite decimals"? That is, given one of the equivalence classes $[\{a_n\}]$ how do we get that
$$
[\{a_n\}] = \sum_{i=-n}^\infty a_i10^{-i}
$$
?

REPLY [2 votes]: Here we map each Cauchy equivalence class $0 \lt [(a_n)] \lt 1$ to its decimal expansion.
Recall the definition of decimal fractions, said another way, the finite decimal expansions. The class $[(a_n)]$ might be represented by a constant sequence with all the terms equal to a decimal fraction. We can then unambiguously associate to this class a terminating decimal expansion.
To handle the remaining Cauchy equivalence classes, we need to prove the following:
Lemma 1: Let $0 \lt [(a_n)] \lt 1$ be given and suppose it can't be represented by a finite decimal expansion. Then for every $k \ge 1$ there exist one and only one decimal fraction
$\tag 1 \frac{h}{10^k} \text{ with } 0 \le h \lt 10^k$
such that for every representative $(b_m)_{\,m \ge 0} \in [(a_n)]$ there exist $N \in \Bbb N$ such that
$\tag 2 \text{For every } n \ge N, \; \frac{h}{10^k} \lt b_n \lt \frac{h+1}{10^k}$
Proof Sketch: Use the triangle inequality over $\Bbb Q$ and the definition of a Cauchy sequence.
Proposition 2: The mapping $k \mapsto h$ gives the first $k$ digits of the infinite decimal expansion to the right of the decimal point, by placing the $k$ digits of the $\text{base-}10$ expansion of the integer $h$ in accordance with the prescription of our positional numeral system
(it may be necessary to 'pad' with zeroes right after the decimal point).

We can directly build the decimal expansion for a real 'Cauchy' number $x$ using the above theory. If $x \lt 0$ we build it for $-x$ and then put the minus sign back.
You can choose any Cauchy sequence representing $x \gt 0$ and you will get the same decimal expansion. Using very simple arguments you 'strip out' the integer part (expansion left of the decimal point) which might actually be $x$. In any case, the problem is reduced to building the expansion for a Cauchy sequence of rationals $(a_n)$ where there exist $N$ such that for $n \ge N$, $\;0 \lt a_n \lt 1$
and $(a_n)$ is not a decimal fraction.
Let $k = 1$. For some $N$ all the terms $a_n$ for $n \ge N$ can be guaranteed to lie in one of the $10$ sub-open-interval of length $\frac{1}{10}$. This '$h$' starts the process of building, from left to right the digits after the decimal point, the decimal estimates for $x$.
You then keep repeating, at each $k$ step dividing each new 'target' into $10$ sub-open-intervals of equal length $10^{-k}$ to get the next decimal digit in $\{0,1,2,\dots,9\}$.
When you get up to the $k^{th}$ build, $d = 0.d_1 d_2 \dots d_k$, we have
$\tag 3  x \gt d \text{ and } x - d \lt \frac{1}{10^k}$