Let $(X,\rho)$ be a metric space, and define $\sigma(x,y)=\min\{1,\rho(x,y)\}$. I have to prove that there are two positive constants $C_1,C_2$ such that
$C_1\rho(x,y)\leq\sigma(x,y)\leq C_2\rho(x,y)$.
Obviously we can take $C_2=1$, but how can we choose $C_1$?