In an ordered field, if all rational Cauchy sequences converge, must all Cauchy sequences converge?

Question

In other words, does there exist an ordered field in which all rational Cauchy sequences converge, but some Cauchy sequences do not converge?

Answer 1

Let $F$ be an ordered field. Then $\mathbb Q$ is a subfield of $F.$

Let $(a_n)_{n=1}^\infty$ be a sequence of members of $\mathbb Q.$ We will call it a Cauchy sequence in $F$ if there is some $\ell\in F$ such that for every positive $\varepsilon\in F,$ all but finitely many $a_n$ are between $\ell\pm\varepsilon.$

Either $F$ is a subfield of $\mathbb R$ or it has some members not in $\mathbb R.$

So we suppose $F$ is not a subfield of $\mathbb R.$ Should I leave it as an exercise to show that in that case $F$ contains some positive members $\iota$ that are smaller than any member of $\mathbb Q\text{?}$ In that case, can any Cauchy sequence in $F$ whose terms are rational converge to $\iota\text{?}$ No rational number is between $\iota\pm\iota/2,$ so it can't happen.

Therefore, the only ordered field in which every Cauchy sequence of rationals converges is $\mathbb R.$