Equivalent Metrics: Convergent Sequences?

[AxiomOfChoice]

Is it true that two metrics d_1 and d_2 on a set X are equivalent iff they have the same convergent sequences (i.e., a sequence that converges in d_1 converges in d_2 and vice versa)?

 

[arkajad]

Yes.

Engelking, "General Topology", Thm. 4.1.2

"Two metrics on X are equivalent iff they induce the same convergence."

 

[nonequilibrium]

Yup, although the correct term is "topologically equivalent", since metrics can also be "uniformly equivalent" or "Lipschitz equivalent" and there might be more such equivalencies.