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!