Ordering on Real Numbers from Decimal Expansion

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Theorem

Let $x, y \in \R$ be real numbers.

Let $x$ and $y$ be expressed by their decimal expansions:


\(\displaystyle x\) \(=\) \(\displaystyle m \cdotp d_1 d_2 d_3 \ldots\)
\(\displaystyle y\) \(=\) \(\displaystyle n \cdotp e_1 e_2 e_3 \ldots\)

Let $\preccurlyeq$ be the lexicographic ordering on $\R$ defined as:

$x \preccurlyeq y$ if and only if:
$m \prec n$
or:
$m = n$ and $\exists k \in \Z_{>0}: \left({\forall j: 1 \le j < k: d_j = e_j}\right) \land d_k < e_k$
or:
$m = n$ and $\forall j \in \Z_{>0}: d_j = e_j$.


Then:

$x \le y$

where $\le$ denotes the usual ordering on $\R$.


Proof


Sources