I know that equivalent metrics do not conserve completeness (e.g $d_1 = |\operatorname{arctan}(x) - \operatorname{arctan}(y)|$ is equivalent to $d_2 = |x - y|$, but $d_1$ does not conserve completeness).
However, equivalent norms do conserve completeness. Does this mean that if two norms are equivalent, their induced metrics are equivalent and they conserve completeness?
