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I need to give an example of two metrics on a set that induce the same topology, but where a sequence is Cauchy relative to one of the metrics and not the other.

Any help would be appreciated! Thanks!

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    Closely related: http://math.stackexchange.com/questions/7578/two-metrics-induce-the-same-topology-but-one-is-complete-and-the-other-isnt2011-04-06

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Hint: What is the topology induced on $\left.\left\{\frac{1}{n}\;\right|\; n\in\mathbb{Z}, n\gt 0\right\}$ by the standard metric?


Alternative example. Take $\mathbb{R}$ with the usual metric, and the metric $d(x,y) = \Bigl|\arctan(x) - \arctan(y)\Bigr|.$ Then consider the sequence $a_n = n$.

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    @caligirl11: Correction: $d(a_n,a_m)=0$ if $n=m$; but for any $N\gt 0$, you can find $n,m\gt N$ with $n\neq m$, and in *that* case you will have $d(a_n,a_m)=1\gt 1/2$. Otherwise, fine.2011-04-06