Order $\leq$ on the real numbers defined via Cauchy sequences

I like to define the real numbers $\mathbb{R}$ as equivalence classes $[(a_n)_{n\geq 0}]$ of Cauchy sequences $(a_n)_{n\geq 0}$ of rational numbers by the equivalence relation: $(a_n)_{n\geq 0}$ is in relation to $(a'_n)_{n\geq 0}$ iff the sequence $(a_n-a'_n)_{n\geq 0}$ converges to zero.

Now, given the order $\leq$ on the rational numbers, I would like to define the total order $\leq$ on $\mathbb{R}$ following this definition. Ideally, I would like to set $[(a_n)_{n\geq 0}] \leq [(a'_n)_{n\geq 0}]$ iff there exists an $N>0$ such that for all $n>N$ we have $a_n\leq a'_n$.

Does this already give me the correct (i.e. the usual) order?

Thanks!