The second-to-last chapter in Spivak's Calculus is called "Construction of the Real Numbers".

The very first chapter of the book, called "Basic Properties of Numbers", Spivak presents twelve properties of numbers.

He says "we have assumed that numbers are familiar objects, and that P1-P12 are merely explicit statements of obvious, well-known properties of numbers. It would be difficult, however, to justify this assumption. Althgouh one learns how to 'work with' numbers in schook, just what numebrs are, remains rather vague. A great deal of this book is devoted to elucidating the concept of numbers, and by the end of the book we will have become quite well acquanited with them. But it will be necessary to work with numbers throught the book. It is therefore reasonable to frankly admit that we do not yet thorougly understand numbers; we may still say that, in whatever way numbers are finally defined, they should certainly have properties P1-P12."

After introducing the concept of limit in chapter 5, and continuity in chapter 6, chapter 7, called "Three Hard Theorems" presents three theorems without showing their proofs. These are left for chapter 8, in which a thirteenth property, the least upper bound property, is introduced. This property allows proofs of the three theorems in chapter 7, and many other previously inaccessible proofs.

I found it interesting that you seem to be saying that this property is actually a theorem in some other context, ie can be proved.

Maybe Spivak will revisit the property in that context once the chapters on sequences and series come up. I will reach them soon!