(Solution) - Remark 2 9 implies that the lexicographic preference relation example 1 114 -(2025 Original AI-Free Solution)

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Remark 2.9 implies that the lexicographic preference relation (example 1.114) cannot be represented by a utility function, since the lexicographic

preference ordering is not continuous. To verify this, assume, to the contrary, that u represents the lexicographic ordering ?L on ?2.
1. For every x1 ? ? there exists a rational number r(x1) such that u(x1; 2) > r(x1) > u(x1; 1)
2. This defines an increasing function r from R to the set Q of rational numbers.
3. Obtain a contradiction.