least upper bound property

The least upper bound property is the single guarantee that the real line has no gaps. It promises that whenever a nonempty set is capped from above, there really is a tightest cap, a number sitting exactly at the edge of the set.

Stated precisely: every nonempty subset of the reals that is bounded above has a supremum, and that supremum is itself a real number. Equivalently, by reflection, every nonempty subset bounded below has an infimum. This is taken as an axiom defining the reals, or proved as a theorem once the reals are constructed.

The power of this single property is enormous. From it flow nearly all the foundational theorems of analysis: the Archimedean property, the Bolzano–Weierstrass theorem, the convergence of bounded monotone sequences, the intermediate and extreme value theorems, and the completeness of the reals as a metric space. Each says, in its own language, that the line has no holes.

The honest contrast is with the rationals, which form an ordered field but lack this property. The rationals less than the square root of 2 are bounded above by rational numbers, yet have no rational supremum; the would-be supremum slips through a gap. Filling all such gaps is exactly what passing from the rationals to the reals accomplishes.