Need to show two metric spaces are equivalent

[Cratylus]

Let X=R

(X,d1) d1(x,y) =|x-y|
(X,d2) d2(x,y)=|[imath] \frac {x}{1-|x|}-\frac {y}{1-|y|}[/imath] |

Here is some work:
Let r>0 Take s=r/(1+|r|)
Consider d2(x,y).
Let y[imath]\in[/imath] B(x,s) then d2(x,y)<s
Then |[imath] \frac {x}{1-|x|}-\frac {y}{1-|y|}[/imath] | <s

I can’t seem to find a value of s to make it work
I tried everything.
any help would be appreciated.
This is not homework

 

[pka]

Let X=R
(X,d1) d1(x,y) =|x-y|
(X,d2) d2(x,y)=|[imath] \frac {x}{1-|x|}-\frac {y}{1-|y|}[/imath] |
Here is some work: Let r>0 Take s=r/(1+|r|), Consider d2(x,y).
Let y[imath]\in[/imath] B(x,s) then d2(x,y)<s Then |[imath] \frac {x}{1-|x|}-\frac {y}{1-|y|}[/imath] | <s

There are two textbooks each of which has a rather good & clear discussion of this topic.
General Topology by Willard and Introduction to topology by Helen Cullen.
Both texts discuss that that two metrics on the same set are equivalent provided each of the metrics generate the same set of basic open sets.