The least upper bound property is a property of the real numbers which distinguishes it from other ordered fields. The property states that every bounded subset of R has a least upper bound in R. The LUBP is an assumed property of the real numbers as it can not be proven from the ordered field axioms and is important because it is essential in the proof that a sequence is Cauchy iff it is convergent (which is to say that the real numbers are a complete ordered field). However, the LUBP can be proven if we assume that a sequence is Cauchy iff it is convergent. Linked is an interesting and fun proof of such, written as an undergraduate.

Proof of the Least Upper Bound Property… but not really