# Is the density of $\mathbb{Q}$ in $\mathbb{R}$ equivalent to the Least Upper Bound Property of $\mathbb{R}$?

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?

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