Definition:Cauchy Equivalent Metrics

Definition
Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$.

That is, $\left({X, d_1}\right)$ and $\left({X, d_2}\right)$ are two different metric spaces on the same underlying set $X$.

Then $d_1$ and $d_2$ are said to be Cauchy equivalent for every sequence $\sequence{x_n}$ of points in $X$:
 * $\sequence{x_n}$ is a Cauchy sequence in $\struct{X,d_1} \iff \sequence {x_n}$ is a is a Cauchy sequence in $\struct{X,d_2}$