\begin{definition}[Definition:Cauchy Equivalent Metrics]
Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$.
That is, $\struct {X, d_1}$ and $\struct {X, d_2}$ are two different metric spaces on the same underlying set $X$.
Then $d_1$ and $d_2$ are said to be '''Cauchy equivalent''' {{iff}} 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 $\struct {X, d_2}$
{{NamedforDef|Augustin Louis Cauchy|cat = Cauchy}}
\end{definition}