I want to find an example of a metric space $X$ with two equivalent (in the sense that all sequences in $(X,d)$ converge to the same limit if and only if all sequences in $(X,e)$ converge to the same limit) metrics such that there exists a sequence which is Cauchy with respect to one of the metrics but not with respect to the other.
I had the idea that if a sequence is convergent then it must be Cauchy. However, it turns out not to be the case (?), if such an example as described above exists. I've been thinking about the "French rail road metric", but no, it didn't seem to do the trick.
Would appreciate a hint.
Update: It was pointed out in the comments below that the statement above wouldn't be true if under equivalence it was meant that two metrics induce the same topology. As far as I know, however, two metrics have sequences convergent to the same limit under two metrics if and only if such metrics are also equivalent in the sense that there exist $m,M>0$ such that $m e(x,y)\le d(x,y)\le M e(x,y), \forall x,y\in X$. Which means that equivalent metrics in the sequence sense are also equivalent in the $m,M$-inequality sense, which means that if topology is preserved under both metrics then such metrics must be equivalent in both senses.
I'm somewhat confused, would appreciate a clarification.