Why isn't the The Least Upper Bound axiom derivable from the order axioms? Apostol, in his Calculus book, has this as one of the axioms before he starts fleshing out rest of the subject. The axiom states that any non-empty set of real numbers that has an upper bound necessarily has a least upper bound. 
I fail to understand why we need to state this as axiom. Isn't this obvious from the order axioms? If a non-empty set has an upper bound, then there must exist the least one. Sorry if I'm unable to articulate it any better, but that could also mean I don't really know what I'm asking. 
The necessity of the Least Upper Bound axiom is not obvious to me. Why is it needed?
 A: It is not obvious from the order axioms because it does not follow from the order axioms. For example, the rationals $\mathbb Q$ with the usual addition and multiplication is a field that satisfies the order axioms but does not have the least upper bound property. For example since $\frac{\pi^2}{6} = \sum_{k=1}^\infty \frac{1}{k^2}$, the set $\{\sum_{k=1}^n \frac{1}{k^2} : n \in \mathbb N\}$ is a bounded subset of $\mathbb Q$ with no least rational upper bound.
A: Think about the rationals in the interval $(-\sqrt{2},\sqrt{2})$. In general rational numbers satisfy the order axioms. However, this set fails to satisfy LUB axiom simply because $\sqrt{2}$ is not a member of rational. 
A: The rational numbers also satisfy the field axioms and the order axioms, but they do not satisfy the Least Upper Bound axiom. For example, consider the set of all rational numbers $q$ such that $q^2 \leq 2$. This is bounded from above, but the least upper bound is $\sqrt{2}$ which is not rational.
The Least Upper Bound Axiom is necessary to make sure that the real line doesn't have any "gaps".
A: The Least Upper property is not a consequence of the partial ordering on the reals. The same ordering exists on the rationals and the LUB property fails for them.For example, the set $\{ x\in\mathbb{Q} : x^2 <2 \} $ does not have a least upper bound in $\mathbb{Q} .$ (Fairly easy to prove,do it yourself.)  The LUB property rests on the completeness of the real numbers, which can be formulated several different ways. What it basically means is that there are no gaps in the real numbers: in between any 2 rational numbers there exists an irrational and in between any 2 irrationals,there's a rational. As a result,all Cauchy sequences converge in the reals and this is what gives the real numbers their unique analytic properties that makes them so paramount in analysis. 
A: If this was true, the rationals would also satisfy it.
In $\mathbb Q$, does the following have a least upper bound / supremum?
$$\{3,3.1,3.14,3.141,3.1415,3.14159,\dots\}$$
