Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 4 / 7F / b / Solution

Suppose $\mathbf{x}⪯\mathbf{y}$. Define the nonnegative tuple $\mathbf{z}$ coordinatewise byWhen $y_i=0$, support inclusion forces $x_i=0$, so $x_i=y_i{z}_i$ in every coordinate.

Conversely, if $x_i=y_i{z}_i$ with $y_i,z_i\ge 0$, then $x_i>0$ implies $y_i>0$. Hence $P(\mathbf{x})\subseteq P(\mathbf{y})$ and $\mathbf{x}⪯\mathbf{y}$. This proves the equivalence.