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!
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!
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 $a_n = n$.