Definition:Cauchy Equivalent Metrics

Definition
Let $d_1: X \times X \to \R_{\ge 0}$ and $d_2: X \times X \to \R_{\ge 0}$ be metrics on a metric space $X$.

Then $d_1$ and $d_2$ are said to be Cauchy equivalent iff every sequence of points in $X$ that is Cauchy under one metric is also Cauchy under the other.