How do we define the order relation in $\mathbb{R}$ in the construction via Cauchy sequences?

Question

I am reading the article of Wikipedia about the contruction of $\mathbb{R}$ via Cauchy sequences, but I don't understand why the order relation that appears in the article is well-defined. It says that for all $[(a_n)],[(b_n)]\in\mathbb{R}$ we have that $[(a_n)]\leq [(b_n)]$ if and only if there exists a natural $N$ such that $a_n\leq b_n$ for all $n\geq N$.

I want to prove that this relation is well-defined.

Let $[(a_n)]$,$[(b_n)]$,$[(c_n)]$ and $[(d_n)]$ be real numbers. Suppose that $[(a_n)]=[(c_n)]$,$[(b_n)]=[(d_n)]$ and $[(a_n)]\leq [(b_n)]$, and our aim is to show that $[(c_n)]\leq [(d_n)]$. However, if we considered $(a_n)=(d_n)=(-\frac{1}{n})$ and $(b_n)=(c_n)=(\frac{1}{n})$, the hypothesis of the statement would hold but the conclusion would fail.

I would thank if you told me where I'm wrong or what I should do.

Answer 1

Yes, you are correct - this is a mistake. The right definition of $[(a_n)]\le[(b_n)]$ is

for every $\epsilon>0$, there is some $N$ such that for all $n>N$ we have $a_n\le b_n+\epsilon$.

(To avoid circularity issues, $\epsilon$ here should be rational. But this is a side point.)

In fact, though, the sort of counterexample you suggest ("two Cauchy sequences approaching the same number from different sides") is not the only way that the proposed definition is problematic. We can for example find two increasing Cauchy sequences which approach the same value but do not satisfy the proposed definition:

Consider e.g. $$0,1.2, 1.21, 1.222,...$$ versus $$1, 1.1, 1.22, 1.221, ...$$ and note that they each converge to $1.22222...$.