Exercise 9, § 5

Let $\Gamma$ be a commutative totally ordered group and $\Gamma^+$ the set of positive elements of $\Gamma$. Denote by (GOD) the following property: (GOD) : Every major subset $M \subset \Gamma^+$ (VI, § 3, no 5, déf. 2) possesses a countable subset, cofinal for the opposite order. a) Now prove that if $\Gamma$ is of height 1, the property (GOD) is satisfied. b) For every integer $n$, now prove that there exists a totally ordered group $\Gamma$ of height $n$ possessing the property (GOD) (take $\Gamma = \mathbf{Z}^n$ endowed with the lexicographic order).