The least upper bound axiom, also abbreviated as the LUB axiom, is an axiom of real analysis. It is an axiom in the sense that it cannot be proven within the system of real analysis. However, like other axioms of classical fields of mathematics, it can be proven from Zermelo-Fraenkel set theory, an external system. The axiom says that if a nonempty subset of the real numbers has an upper bound, then it has a least upper bound. This axiom is very useful since it is essential to the proof that the real number line is a complete metric space. The rational number line does not satisfy the LUB axiom and hence is not complete. A perfect example is . 2 is certainly an upper bound for the set. However, this set has no upper bound — for any , we can find a with y > x. In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ... Please refer to Real vs. ... In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S. The term lower bound is defined dually. ... In mathematics, the supremum of an ordered set S is the least element (not necessarily in S) which is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

Proof that the real number line is complete

Let be a Cauchy sequence. Let S be the set of real numbers that are bigger than s_n for only finitely many . Let . Let be such that . So, the sequence passes through the interval infinitely many times and through its complement at most a finite number of times. That means that and hence . Clearly, is an upper bound for S. By the LUB Axiom, let b be the least upper bound. . By the triangle inequality, . Therefore, and so is complete. Q.E.D. In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the triangle inequality states that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, Lp spaces (p ≥ 1) and in all inner product spaces... Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek (hóper édei deĩxai) which was used by many early mathematicians including Euclid and Archimedes. ...