In the process of touching up some notes on infinite series, I came across the following "result":
Theorem: For an ordered field $(F,<)$, the following are equivalent:
(i) Every Cauchy sequence in $F$ is convergent.
(ii) Absolutely convergent series converge: $\sum_n |a_n|$ converges in $F$ $\implies$ $\sum_n a_n$ converges in $F$.
But at present only the proof of (i) $\implies$ (ii) is included, and unfortunately I can no longer remember what I had in mind for the converse direction. After thinking it over for a bit, I wonder if I was confusing it with this result:
Proposition: In a normed abelian group $(A,+,|\cdot|)$, the following are equivalent:
(i) Every Cauchy sequence is convergent.
(ii) Absolutely convergent series converge: $\sum_n |a_n|$ converges in $\mathbb{R}$ $\implies$ $\sum_n a_n$ converges in $A$.
For instance, one can use a telescoping sum argument, as is done in the case of normed linear spaces over $\mathbb{R}$ in (VIII) of this note.
But the desired result is not a special case of this, because by definition the norm on a normed abelian group takes values in $\mathbb{R}^{\geq 0}$, whereas the absolute value on an ordered field $F$ takes values in $F^{\geq 0}$.
I can show (ii) $\implies$ (i) of the Theorem for ordered subfields of $\mathbb{R}$. Namely, every real number $\alpha$ admits a signed binary expansion $\alpha = \sum_{n = N_0}^{\infty} \frac{\epsilon_n}{2^n}$, with $N_0 \in \mathbb{Z}$ and $\epsilon_n \in \{ \pm 1\}$, and the associated "absolute series" is $\sum_{n=N_0}^{\infty} \frac{1}{2^n} = 2^{1-N_0}$.
Because an ordered field is isomorphic to an ordered subfield of $\mathbb{R}$ iff it is Archimedean, this actually proves (ii) $\implies$ (i) for Archimedean ordered fields. But on the one hand I would prefer a proof of this that does not use the (nontrivial) result of the previous sentence, and on the other hand...what about non-Archimedean ordered fields?
Added: The article based on this question and answer has at last appeared:
Clark, Pete L.; Diepeveen, Niels J.; Absolute Convergence in Ordered Fields. Amer. Math. Monthly 121 (2014), no. 10, 909–916.
If you are a member of the MAA, you will be frustrated if you try to access it directly: the issue is currently advertised on their website but the articles are not actually available to members. The article is available on JSTOR and through MathSciNet. Anyway, here is an isomorphic copy. Thanks again to Niels Diepeveen!