1
$\begingroup$

Consider the following two assertions:

  • (Least Upper Bound Property) If $S$ is a subset of $\mathbb{R}$ which has an upper bound, then $S$ has a least upper bound.
  • (Density of $\mathbb{Q}$ in $\mathbb{R}$) If $x,y\in\mathbb{R}$, then there is $q\in\mathbb{Q}$ such that $x<q<y$.

Are these two statements equivalent (assuming that $\mathbb{R}$ is an ordered field)? What if we assume that $\mathbb{R}$ is an archimedean ordered field? If so, can you point me to a reference where I can find a proof?

$\endgroup$
  • $\begingroup$ You're right, of course. (Now I feel stupid for not seeing the obvious counter-example.) $\endgroup$ – user355272 Jul 20 '16 at 17:34
  • $\begingroup$ Maybe you were thinking of something more along the lines of Cauchy Completeness, which is equivalent. $\endgroup$ – Andres Mejia Jul 20 '16 at 17:42
2
$\begingroup$

(comment turned answer) $\mathbb{Q}$ is an ordered archimedean field which has $\mathbb{Q}$ dense in it, but doesn't have LUB property, so this is not enough.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.