Equivalence of d and p Metrics on X

[hypermonkey2]

Homework Statement

Show that d and p are equivalent metrics on X where p=d(x,y)/(1+d(x,y))

Homework Equations

ive proved already that p is indeed a metric too (if d is a metric).

The Attempt at a Solution

I believe I am supposed to use the Lipschitz condition where there exits constants A and B st for all x,y,

Ap<=d<=Bp

but i think i can actually prove that one of these two constants cannot exist... and i using the wrong definitions? Thanks!

 

[Dick]

Two metrics are equivalent if they induce the same topology. Isn't that the real definition? It is true that Ap<=d<=Bp shows that. But isn't it also true that for a metric topology what really determines the topology is the 'small' sets? If you can show there is a condition like that for say d<=1. That would also suffice, it doesn't have to hold for ALL x,y. Just for nearby ones.