Ordering on Real Numbers from Decimal Expansion

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

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

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


 * $x \preccurlyeq y$ :
 * $m \prec n$
 * 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 $\R$.