Definition:Equivalent Metrics
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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 $\struct {X, d_1}$ if and only if $x_n \to x$ in $\struct {X, d_2}$.
Then $d_1$ and $d_2$ are equivalent metrics.