Metric Space Open Sets. Let $(X, \rho)$ be a metric. I've shown $\sigma(s,t) = \frac{\rho(s,t)}{1 + \rho(s,t)}$ is also a metric on $X$. 
I'm having trouble showing that the open sets defined by the metric $\rho$ are the same as the open sets defined by $\sigma$. I know I must show that an open ball in the $\rho$ metric is an open set in the $\sigma$ metric, and that an open ball in the $\sigma$ metric is an open set in the $\rho$ metric. Any hints or advice? 

Hint: note that $\sigma=\frac{\rho}{1+\rho}=1-\frac{1}{1+\rho}$ is strictly increasing on $[0;+\infty)$, therefore, $\sigma&lt\epsilon$ implies $\rho&lt\delta$ (for some $\delta$) and vice versa.
