Show that for any two Cauchy sequences of rational numbers, their difference is a equivalent to a sequence of nonnegative numbers Let $a_n$ and $b_n$ be Cauchy sequences of rational numbers, either $b_n-a_n$ or $a_n-b_n$ is a sequence of nonnegative numbers.
I don't really understand how this is true, I think first we have to assum $a_n\neq b_n$. If we assume that they are not the same sequence, how does this work? 
 A: You don't have to assume that the sequences are not the same; if they are then $b_n - a_n$ is the zero sequence, which is nonnegative.
Let $x$ be the real number represented by the Cauchy sequence $b_n - a_n$. Then either $b_n - a_n$ or $a_n - b_n$ represents the nonnegative real number $|x|$. Recall that Cauchy sequences are equivalent when they represent the same real number. So you have to find a nonnegative Cauchy sequence that represents the real number $|x|$.
A: We don't have to assume $a_n\ne b_n$.  Let $c_n=a_n-b_n$.  If $a_n=b_n$ for all $n$, then $c_n=0$ for all $n$.  Remember that non-negative doesn't mean positive, it means $\ge 0$.
If $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences, then they both converge, let
$$\lim_{n\to\infty}a_n=1\quad\text{and}\quad\lim_{n\to\infty}b_n=b.$$  Now, either
$a\le b$ or $b\le a.$
Suppose $a\ge b$, then $\lim_{n\to\infty}(a_n-b_n)=a-b\ge 0$.  So, what can you say about $a_n-b_n$ if $n$ is large enough?
A: This sentence with

either $b_n-a_n$ or $a_n-b_n$ is a sequence of nonnegative numbers

is not true in this form: you can modify the first some elements of a sequence anyhow without affecting its being Cauchy. What is true, is either


*

*one of the sequences $b_n-a_n$ and $a_n-b_n$ is equivalent to (=has the same limit as) a Cauchy sequence of nonnegative (rational) numbers.


as Arthur answered, or


*there is an index $N$ such that one of $a_n-b_n$ and $b_n-a_n$ is nonnegative for all $n>N$.

