Example of the two uniformly equivalent metrics, one is bounded while another is not.

Question

Can anyone gives me an example such that two metric space (X,d1), (X,d2) are uniformly equivalent. And the metric space (X,d1) is bounded and metric space (X,d2) is not bounded?

Answer 1

$(\Bbb R, \min\{|x - y|,1\})$ and $(\Bbb R, |x - y|)$.

I'll leave it to you to show that the first is indeed a metric. The equivalence should be obvious. The same trick works for converting any unbounded metric on any metric space to a bounded one.