TITLE: Can a Cauchy sequence converge for one metric while not converging for another?
QUESTION [12 upvotes]: Is there an easy example of one and the same space $X$ with two different metrics $d$ and $e$ such that one and the same sequence $\{x_n\}$ is a Cauchy sequence for both metrics, but converges only for one of them?

REPLY [8 votes]: Having "the same X" is somewhat meaningless, since there's no shared structure other than cardinality. Given any other set Y with metric $d_Y$ and a injective map $\phi: X \rightarrow Y$, you can define $e(x_1,x_2) = d_Y(\phi(x_1),\phi(x_2))$. So all you have to do is find two spaces with the same cardinality where one has a convergent Cauchy sequence and the other has a non-convergent one. Or find a convergent Cauchy sequence and a non-convergent Cauchy sequence in the same space, then map one sequence to the other.