Definition:Cauchy Equivalent Metrics

Let $$d_1: X \times X \to \R_+$$ and $$d_2: X \times X \to \R_+$$ 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.