Let $R$ be an Archimedean ordered field, and $S$ a non-Archimedean extension of $R$. Then $R$ is complete if and only if $S$ admits a standard part; namely, every limited element of $S$ is infinitely close to an element of $R$.

Here an element of $S$ is limited if it is between two elements of $R$. Two elements of $S$ are infinitely close if the difference is an infinitesimal, and an infinitesimal is between $-r$ and $r$ for every positive element $r\in R$.

Let $R$ be an Archimedean ordered field, and $S$ a non-trivial ultrapower extension of $R$. Then $R$ is complete if and only if $S$ admits a standard part; namely, every limited element of $S$ is infinitely close to an element of $R$. Here an element of $S$ is limited if it is between two elements of $R$. Two elements of $S$ are infinitely close if the difference is an infinitesimal, and an infinitesimal is between $-r$ and $r$ for every positive element $r\in R$.

A similar construction can be carried out in SPOT without ultrafilters.