Prove that a sequence is Cauchy in (X,d) if and only if it is Cauchy in (X,d')

Question

We have that d and d' are strongly equivalent metrics on a set X. I want to prove that a sequence is Cauchy in (X,d) if and only if it is Cauchy in (X,d'). Just wondering how to start this problem

Answer 1

We have, with $a,b>0$, that

$ad(x,y) \le d'(x,y) \le bd(x,y)$ for all $x,y \in X$.

Now let $(x_n)$ be a sequence in $X$. Then we get

$ad(x_n,x_m) \le d'(x_n,x_m) \le bd(x_n,x_m)$ for all $n,m \in \mathbb N$ .

Can you take it from here ?