Chapter 7.1, Problem 22E

Prove that if $F$ is an ordered field with $F^{+}$ as its set of positive elements, then $F^{+}\supseteq \left\{ne\right|n\in {\mathbb{Z}}^{+}\}$, where $e$ denotes the multiplicative identity in $F$. (Hint: See Theorem 5.34 and its proof.)

Theorem 5.34: Well-Ordered $D^{+}$

If $D$ is an ordered integral domain in which the set $D^{+}$ of positive elements is well-ordered, then

$e$ is the least element of $D^{+}$ and

$D^{+}=\left\{ne\right|n\in {\mathbb{Z}}^{+}\}$.