This question has already been asked and answered here
Let $ϵ>0$ be given. With loss of generality, we may assume $ϵ$ is rational. Suppose $a_n$ is a Cauchy sequence and $b_n, a_n$ are equivalent.
Choose N such that $|b_k−a_k|<\frac13\epsilon$ and $|a_n−a_m|<\frac13\epsilon$ whenever $k,m,n≥N.$ We can do this because an is Cauchy, and $a_n,b_n$ are equivalent.
Then if $n,m≥N$, we have
$|b_n−b_m|≤|b_n−a_n|+|a_n−a_m|+|a_m−b_m|≤\frac13\epsilon+\frac13\epsilon+\frac13\epsilon=ϵ.$ Hence $b_n$ is Cauchy.
Reversing the roles of $a_n,b_n$ finishes the proof.
What I cannot follow in the answer is why we can assume:
Am I missing something huge here or I am out of the loop on some standard inequality?