 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ia/paper-4/7f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Suppose

x ⪯ y

. Define the nonnegative tuple

z

coordinatewise by

z_{i} = {x_{i} / y_{i}, 0, y_{i} > 0, y_{i} = 0.

When

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} \geq 0

, then

x_{i} > 0

implies

y_{i} > 0

. Hence

P (x) \subseteq P (y)

and

x ⪯ y

. This proves the equivalence.

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