Definition:Cauchy Equivalent Metrics

Let $$d_1:X \times X \to \mathbb{R}_+$$ and $$d_2:X \times X \to \mathbb{R}_+$$ be metrics on a metric space $$X \ $$.

These two metrics 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.