if (X p) is compact, and (X, d) is equivalent, then the map id: X -> X is a homeomorphism between the two topology spaces. So (X, d) is also compact, which imples (X, d) is bounded.
I have question about compact metric space. I know that every compact metric space is bounded since compactness implies totally boundedness and totally boundedness implies boundedness. But is it true that if (X,p) is a compact metric space, then every equivalent metric d of p on X is also bounded? I think it is true, but have no clue how to prove it.