= Finite-support Hahn embedding along a countable convex chain
{title2=$G\cong\bigoplus_{n\ge1}\mathbb R$}
Let a divisible ordered <Abelian group> be the union of a strictly increasing sequence of convex subgroups $C_1=0,C_2,\ldots$, with each ordered quotient $C_{n+1}/C_n$ isomorphic to the additive real line. Convexity makes every $C_n$ divisible, so these are rational <vector subspaces>. Choose a rational-linear section for each quotient. Every element then has a unique finite sum of section values. Its sign is the sign of the coefficient of the largest active index, because convexity makes the corresponding ordered quotient decide its sign. The coefficients give an embedding, indeed an ordered-group isomorphism, into the <Hahn group> with positive integer indices in the maximum-support convention. No compatibility of a separate real scalar multiplication is needed for this group embedding.
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