I am trying to prove the existence of a standard map for $\overline{\mathbb Q}=\mathrm{fin}(^*\mathbb Q)/\mathrm{inf}(^*\mathbb Q)$ with $\mathrm{fin}(^*\mathbb Q)$ the set of finite numbers of $^*\mathbb Q$ and $\mathrm{inf}(^*\mathbb Q)$ the set of its infinitesimal numbers.
In other words, I am trying to prove that $\forall y\in\mathrm{fin}(^*\overline{\mathbb Q}),\exists x\in\overline{\mathbb Q},y\sim\ ^*x$.
Probably the most straighforward proof is to use the isomorphism of $\overline{\mathbb Q}$ with $\mathbb R$ obtained by sending an element of fin$({}^\ast\mathbb Q)$ to its standard part, which is an element of $\mathbb R$. Elements of $\overline{\mathbb Q}$ differing by an infinitesimal in inf$({}^\ast\mathbb Q)$ obviously have the same standard part and therefore the isomorphism is well-defined.