
19.1.1.12 Properties of metric spaces
Condition: Let \( \rho(x, y)- \) be a metric on the set \( X \). Prove that the functions \[ \begin{array}{l} \rho_{1}(x, y)=\frac{\rho(x, y)}{1+\rho(x, y)} \\ \rho_{2}(x, y)=\ln (1+\rho(x, y)), \\ \rho_{3}=\min \{1, \rho(x, y)\} \end{array} \] are also metrics.