I'm not sure how to show that every proper elementary extension of the real numbers has an infinitesimal element. I've been trying to somehow apply the property
$$\forall x \forall y (((\forall \varepsilon > 0)(\lvert x – y \rvert < \varepsilon)) \leftrightarrow x = y)$$
because it seemed useful, although I have no better reason to believe this. If a nonstandard model has no infinitesimal element, then $(\forall \varepsilon > 0)(\lvert x – y \rvert < \varepsilon)$ is equivalent to $(\forall \varepsilon \in \mathbb{R}^+) (\lvert x – y \rvert < \varepsilon)$ where $\mathbb{R}$ is the natural embedding of the standard model, so perhaps this is somehow useful. Unsure how to proceed.
Best Answer
Let $R$ be a proper elementary extension of $\mathbb{R}$, and $r\in R\setminus \mathbb{R}$.