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?

  • $\begingroup$ 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

(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.


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