I want to come up with an example which shows that there exists a sequence which is Cauchy under both metrics, say, $d$ and $e$, in the same space $X$, yet these metrics are not equivalent.
Since we know that if two metrics have the same convergent sequences then they are equivalent, I need to come up with an incomplete metric space. I've been thinking for quite a while but coulnd't come up with such $X$ and $d$ and $e$. One idea is to think of such a metric which will give $0$ for some two points, whereas the other metric will give some number greater than zero for the same two points in $X$. But what could that metric be?
Would appreciate a hint.
A less trivial example than taking constant sequences, is to take a sequence that converges in two non-equivalent metrics on a set. (Just having one such sequence is easy enough), e.g. $\frac{1}{n}e_n$ in $\ell_2$ in its $\ell_2$ metric and in the metric it inherits from $\mathbb{R}^\mathbb{N}$ in the product metric. In both metrics it converges to the $0$ sequence, and so is Cauchy in both of them.