Equivalence Class and Cauchy Let A be an equivalence class in ˆQ different from the class of null sequences P.
P: null sequences
{Q}: the set of all Cauchy sequences in (Q,|·|)
ˆQ={Q}/P
Prove that there exists a representative Cauchy sequence {an}∈ A and a constant c>0 such that either for all n an>c, or an<−c.
I'm having trouble starting this problem and can't figure out the proof.
 A: Cauchy sequences are the ones that 'want to' converge somewhere. In the case of $\Bbb Q$, its Cauchy sequences are exactly the ones that converge in $\Bbb R$. 
Moreover, two Cauchy sequences are regarded the same in ${}^\land Q$ iff they differ in a null sequence, i.e. iff they (want to) converge to the same real number.
Now, take any Cauchy sequence $(a_n)$ which is not a null sequence. Then there is $\varepsilon>0$ such that $|a_n|>\varepsilon$ infinitely many times. It implies that either $a_n>\varepsilon$ or $a_n<-\varepsilon$ holds for infinitely many $n$. Without loss of generality, we suppose the former.
Secondly, apply the Cauchy property for $c:=\varepsilon/2$ to obtain an infinite tail of $(a_n)$ above $c$, i.e. $a_n>c$ when $n>N$ for some $N\in\Bbb N$.
Finally, change the head $(a_n)_{n<N}$ arbitrarily above $c$.
Note that $\varepsilon$ hence $c$ can be chosen as rational, hence we can talk about / prove things about real numbers using only rational numnbers and rational sequences.
