Definition:Equivalent Metrics

Definition
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 set $X$.

Let $\sequence {x_n}$ be a sequence in $X$.

Let $n \to \infty$.

Suppose that $x_n \to x$ in $\struct {X, d_1}$ $x_n \to x$ in $\struct {X, d_2}$.

Then $d_1$ and $d_2$ are equivalent metrics.