Spivak Calculus — Postulate 13 I don't find Postulate 13 of Spivak's Calculus trivial, nor can I understand why it's true.
Postulate 13: Every non-empty set of real numbers that is bounded above has a least upper bound (sup).
Why is this postulate true? Any proof/intuition behind it?
Edit: Let me pose a few questions.

*

*Suppose I decide to devise a pathological function $f$ : $\mathbb{R}$ -> $\mathbb{R}$ which is bounded above but has no supremum. Why will I fail to find such $f$? If you simply take it as an axiom, there's no guarantee I won't be successful.


*Suppose $S$ is an arbitrary non-empty set of real numbers that is bounded above. Does there exist an algorithm to determine $sup(S)$?

This is the completeness axiom, in other words, the Completeness Axiom guarantees that, for any nonempty set of real numbers $S$ that is bounded above, a sup
exists (in contrast to the max, which may or may not exist (see the examples above).
An analogous property holds for inf S: Any nonempty subset of $\mathbb{R}$ that is bounded below has a greatest lower bound
