Define $\displaystyle \rho$ on $\displaystyle X \times X$ by

$\displaystyle \rho(x,y) = min(1,d(x,y)), \ \ \ \ \ \ \ \ \ \ x,y \in X$

Show that $\displaystyle \rho$ is a metric that is equivalent to d

My solution:

Two metrics are equivalent if and only if the convergent sequences in $\displaystyle (X,d)$ are the same as the convergent sequences in $\displaystyle (X,\rho)$

Let $\displaystyle (x_{n})^\infty_{n=1}$ be convergent in $\displaystyle \rho$ i.e.,

$\displaystyle \forall \ \epsilon \ \exists \ N_{\epsilon}\ \mbox{such that}\ \forall\ n \geq N_{\epsilon}\ \rho(x_n,x)={min(1,d(x_{n},x))}}<\epsilon $

Choose $\displaystyle 0 \ < \epsilon \ < 1$. Then $\displaystyle \exists \ N_{\epsilon}$ such that $\displaystyle \forall\ n \ \geq N_{\epsilon}: min (1,d(x_n,x)) < \epsilon < 1$.

This implies $\displaystyle min (1,d(x_n,x))=d(x_n,x)\ <\ \epsilon$.

What if $\displaystyle \epsilon > 1$?