Definition:Equivalent Metrics

Definition
Let $X$ be a set upon which there are two metrics $d_1$ and $d_2$.

That is, $\left({X, d_1}\right)$ and $\left({X, d_2}\right)$ are two different metric spaces on the same set $X$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $X$.

Let $n \to \infty$.

Suppose that $x_n \to x$ in $\left({X, d_1}\right)$ iff $x_n \to x$ in $\left({X, d_2}\right)$.

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