Show that a sequence is Cauchy in $d_1$ iff it's Cauchy in $d_2$

Question

Suppose two distances on $M$ are strongly equivalent, meaning that there exists $k$ and $K$ such that $$d_1(x,y) \le k*d_2(x,y); d_2(x,y) \le K*d_1(x,y)$$.Show that a sequence in $(M, d_1)$ is Cauchy iff it is Cauchy in $(M, d_2)$.

I'm having trouble grasping how to prove this. Let's say $d_1(x_m, x_n) \le r$ for some $m, n \ge N$ (given by definition of Cauchy). Do I have to show that this holds for $d_2$ as well? If so I am not sure how to do this.

Any help is appreciated. Thank you very much!

Answer 1

$d_2(x_m,x_n)\leq Kd_1(x_m,x_n)$. If you want it to be smaller than $r$ then pick $N\in\mathbb{N}$ such that for all $m,n\geq N$ we have $d_1(x_m,x_n)\leq\frac{r}{K}$.