Same open sets + same bounded sets => same Cauchy sequences?

[lugita15]

Let $d_1$ and $d_2$ be two metrics on the same set $X$. Suppose that a set is open with respect to $d_1$ if and only if it is open with respect to $d_2$, and a set is bounded with respect to $d_1$ it and only if it is bounded with respect to $d_2$. (In technical language, $d_1$ and $d_2$ induce the same topology and the same bornology.) My question is, does this imply that a sequence is Cauchy with respect to $d_1$ if and only it is Cauchy with respect to $d_2$?

If not, does anyone know of an example of two metrics which share the same open sets and the same bounded sets, but have different collections of Cauchy sequences?

 

[mathman]

Set of all reciprocal of integers between 0 and 1. Let $d_1$ be defined as $d_1(x,y)=1$ if $x\ne y$. Let $d_2$ be defined as $d_2(x,y)=|x-y|$ if $x\ne y$. Then $(\frac{1}{2},\frac{1}{3},...)$ will be a Cauchy sequence for $d_2$, but not for $d_1$..

 

[lugita15]

Thanks, that makes sense.