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? 

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 $n>N$.

