elementary number theory - Definition of real by infinite series instead of their Cauchy limits

Looking at Wikipedia´s definition of real numbers I choose a variant one of the alternative definitions, using Cauchy limits. However, Instead of taking a limit I choose the number to be represented by an infinite expansion in a base (here taken as 10), that is an infinite set of numbers 0-9, and I simplicity I take numbers between, and including, 0 and 1. I haven´t used the term "set" but I realize my numbers may be taken to be that.

I now state that $a≡b$ if $0.a_1 a_2 a_3 a_4 a_5…≡0.b_1 b_2 b_3 b_4 b_5…$ if $a_n=b_n$ for all n.

Furthermore, I have $a>b$ if the first $a_n>b_n$ in a´s and b´s expansion respectively including the first number 0 or 1.
For addition and multiplication I establish the infinite series by starting from the highest decimal just as one might do for final numbers. That is, the generating process is similar to addition and multiplication of infinite series such as Euler first did, solving the “Basel problem” of the sum of $(1/n)^2$. (I understand it was perhaps not considered rigorous but I believe it is when the corresponding series converge).

The rules imply that there is a zero corresponding to the expansion $0.00000…$ and that

$1.00000…>0.99999…$

and

$1.00000…-0.99999…=0.00000...$

This is agreeable since it brings to mind the intuitive (or perhaps naive) idea from the number line, that numbers are actually “touching” each other. Another issue is that I cannot see a process to “get away” from $1.00000…$ to any “next” or other near number since $1-(0.99999…)^n≡0$ for all n. It therefore brings on the problem that for a not identical to b, I still have $a-b≡0$.

In Wikipedia´s article on Peano axioms I couldn´t find the sign $>$, only $≥$ (except in the section “Equivalent axiomatizations”) but I take it that a-b cannot be 0 if a>b (and that is perhaps how $>$ is defined).

I can introduce “=” or “almost equal” so that a and b are “almost equal” if $a-
b≡0$ but I believe that would take me back to the normal definition of real numbers using Cauchy limits.

If I go down the road where $1.00000…>0.99999…$ all my “new real numbers” would be strictly ordered and I could add and multiply.
Is there any difference (other than the unusual use of $<$) between this and the real numbers defined in Wikipedia?