What makes R different than Q so that R possesses least upper bound property. Q is also an ordered field.
Definition :
An ordered field $(F, +, ·, ≤)$ is said to have the least upper bound property provided that any non-empty subset $A$ of $F$ that is bounded above has a least upper bound.
Lets take $(0,1)$ isn't 1 an upper bound? in $Q$
Best Answer
Even though ${\mathbb Q}$ is "dense in itself" it becomes evident in mathematical praxis that there are important numbers missing that we need, and are convinced "to be there", and which are not available in ${\mathbb Q}$. It's not just special values like $\sqrt{2}$, or $e$ and $\pi$, but in fact any infinite decimal which is not ultimately periodic, and there are many of these. The completion process invented by Dedekind fills these fissures with a sort of "glue" so that in the end a perfect system results, which is not only "order complete", but allows of all the operations which we are so fond of from primary school arithmetic of fractions.