Convex subgroups of a finite-rank ordered Abelian group ($\#\{\text{convex subgroups}\}\le r+1$)

For a torsion-free abelian group of finite rational rank $r$, each convex subgroup of an ordered group is pure: if $ng\in C$ for a positive integer $n$, then $g$ lies between zero and $ng$ or their negatives and hence belongs to $C$. A proper inclusion of convex subgroups therefore increases rational rank. They form a chain, so there are at most $r+1$, including zero and the whole group. A lexicographically ordered copy of ${\mathbb{Z}}^r$ attains the bound.