Some questions about the proof of the properties of the uniform equivalent.

Question

I want to prove some properties of uniformly equivalent metrics. Suppose (X,d) and (X,p) are uniformly equivalent, then the identity map and its inverse are uniformly continuous.

1. The former is bounded does not mean the latter is bounded.

Could anyone please gives me some clues on how to prove these statements? I wanna to give a counterexample to show that the former is bounded does not mean the latter is bounded. But I'm stuck here. Appreciate any suggestions.

Answer 1

Take two different discrete metrics on an infinite set, e.g. the usual metric on $\Bbb Z$ and the metric $\rho(x,y)=1$ if $x\ne y$.