Definition:Equivalent Metrics

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.