Hi, I'd be greatful for any help on the following problem; I have to show that two metrics $\displaystyle \rho$ and $\displaystyle \sigma$ are equivalent if and only if every open ball $\displaystyle B_s^{\sigma}(x)$ contains an open ball $\displaystyle B_r^{\rho}(x)$ and every open ball $\displaystyle B_r^{\rho}(x)$ contains an open ball $\displaystyle B_s^{\sigma}(x)$.

Here is my proof thus far:

If $\displaystyle \rho$ and $\displaystyle \sigma$ are equivalent, then we have that $\displaystyle c\rho(x,y) \leq \sigma(x,y) \leq C\rho(x,y)$. Now, taking $\displaystyle \sigma(x,y) \leq C\rho(x,y)$...

Here is where I get stuck - I want to say the following:

...for large enough s, we can say that $\displaystyle \sigma(x,y) < C\rho(x,y) < s$, and so we have that $\displaystyle \sigma(x,y) < s$ is indeed our set $\displaystyle B_s^{\sigma}(x)$. Taking $\displaystyle C\rho(x,y) < s$, and setting $\displaystyle r = \frac{s}{C}$, we also have our set $\displaystyle B_r^{\rho}(x)$, and so we conclude that $\displaystyle B_s^{\sigma}(x) \subset B_r^{\rho}(x)$.

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!