Subfield of $\mathbb{R}$ that satisfies the Ratio Test must contain every reals.

Question

I am reading James Propp's Real Analysis in reverse. There in pg. 12 he shows Ratio Test $\implies$ LUB Axiom.

1. First he shows that Ratio Test $\implies$ Archimedean Property:
   By Consider the following sequence $\left(\frac{1}{2^n}\right)$ then $\left(|\frac{a_{n+1}}{a_n}|\right) \to \frac{1}{2}<1$.
   So $\sum\limits_{n=1}^\infty \frac{1}{2^n}$ is convergent. i.e the sequence of partial sums $\left(\frac{1}{2},\frac{3}{4},\frac{7}{8},...\right)$ is convergent. If the field is non archimedean then $\exists \varepsilon(>0) \in \mathbb{F}$ with $\varepsilon<\frac1n$ for all $n \in \mathbb{N}_{\mathbb{F}}$.
   Now for any $n\in \mathbb{N}$, $\varepsilon<|s_{n+1}-s_n|=\frac12$ i.e $(s_n)$ is not cauchy, contradiction.
   So $\mathbb{F}$ must be Archimedian.

2. Then he explains that Every Archimedean Ordered Field can be embedded in $\mathbb{R}$.
   I have tried to write a proof for this here.

3. Then he claims Subfield of $\mathbb{R}$ that satisfies the Ratio Test must contain every reals.
   He says it suffices to show that every real number can be written as a sum $n\pm \frac12\pm\frac14\pm\frac18\pm ... $ which satisfies the hypothesis of the Ratio Test.

My questions:

1. The claim of 3 is intuitively clear to me. But I am unable to give rigorous prove.