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Let a divisible ordered Abelian group be the union of a strictly increasing sequence of convex subgroups $C_1=0,C_2,\dots$, 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.

The additive Abelian group $G$ is divisible because it is a real vector space. Each convex subgroup $C_n$ is divisible too: for $c\ge 0$ in $C_n$ and a positive integer $m$, its unique additive quotient $c/m$ satisfies $0\le c/m\le c$, hence belongs to $C_n$. Negative elements follow by taking inverses. Thus all $C_n$ are rational vector subspaces, independently of whether the given real scalar multiplication respects the order.

Choose an ordered-group isomorphism ${\theta }_n:C_{n+1}/C_n\to \mathbb{R}$. It is rational-linear, since additive homomorphisms between divisible torsion-free groups preserve division by integers. Extend a rational basis of $C_n$ to one of $C_{n+1}$; equivalently choose a rational-linear section $s_n:\mathbb{R}\to C_{n+1}$ of ${\theta }_n$ composed with the quotient map. ThenSince $C_1=0$ and the union is $G$, each $g\in G$ has a unique expression $g={\sum }_{j=1}^N{s}_j(r_j)$ with only finitely many nonzero coefficients. Define $\Phi (g)=(r_1,r_2,\dots )$. This is an injective additive group homomorphism. If $r_N\ne 0$ is the largest active coefficient, the image of $g$ in the ordered quotient $C_{N+1}/C_N$ has sign equal to that of $r_N$. Convexity ensures that this is also the sign of $g$.

The maximum-support convention for the Hahn group $V({\mathbb{Z}}_{>0},\mathbb{R})$ uses precisely finite supports and the sign of the largest active coefficient. Hence $\Phi$ preserves and reflects the total order, and therefore preserves join and meet. This proves the finite-support Hahn embedding along a countable convex chain. In fact every finite coefficient sequence is obtained, so $\Phi$ is an ordered-group isomorphism in this convention; the requested embedding follows.