19.1.1.1 Properties of metric spaces

Problem: Is \( \rho(x, y)=\max _{1 \leq k \leq n} k\left(x_{k}-y_{k}\right) \) a metric on \( \mathbb{R}^{n} \), where \( x, y \in \mathbb{R}^{n}, x=\left(x_{1}, \ldots, x_{n}\right), y=\left(y_{1}, \ldots, y_{n}\right) \) ?