Hi, I'd be greatful for any help on the following problem; I have to show that two metrics and are equivalent if and only if every open ball contains an open ball and every open ball contains an open ball .
Here is my proof thus far:
If and are equivalent, then we have that . Now, taking ...
Here is where I get stuck - I want to say the following:
...for large enough s, we can say that , and so we have that is indeed our set . Taking , and setting , we also have our set , and so we conclude that .
I am not allowed to use any arguments using cauchy sequences - only the equivalence inequality relation used above. I'd be so grateful for any help provided!