I have seen elsewhere that the way to do this is to assume you have two sequences $a_n$ and $b_n$, assume $a_n$ is Cauchy and $a_n$ and $b_n$ are Cauchy. Then Let $\epsilon >0$ Suppose $(a_n)$ is Cauchy and $b_n$, $a_n$ are equivalent. Suppose $K\in \mathbb N$, there's come $M\in \mathbb N$ such that $\lvert b_k-a_k\rvert < \frac{\epsilon}{3}$ and $\lvert a_n-a_p\rvert < \frac{\epsilon}{3}$ where $n, p, k \geq M$. We do this because $a_n$ is Cauchy and $a_n$, $b_n$ are equivalent. Then if $n,p,k \geq M$, we have $\lvert b_n-b_p\rvert \leq \lvert b_n-a_n\rvert + \lvert a_n-a_p\rvert + \lvert a_p-b_p\rvert \leq \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3} = \epsilon$. Hence $b_n$ is Cauchy.
Is this the correct approach?