Least upper bound axiom

The least upper bound axiom, also abbreviated as the LUB axiom, is an axiom of real analysis stating that if a nonempty subset of the real numbers has an upper bound, then it has a least upper bound. 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. 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.

An example is S = { xin mathbb{Q}|x^2 < 2}. 2 is certainly an upper bound for the set. However, this set has no least upper bound &mdash; for any upper bound x in mathbb{Q} , we can find another upper bound y in mathbb{Q} with y < x.

Proof that the real number line is complete

Let { s_n}_{ninN} be a Cauchy sequence. Let S be the set of real numbers that are bigger than s_n for only finitely many ninN. Let varepsiloninR ^+. Let NinN be such that forall n,mge N, |s_n-s_m|. So, the sequence passes through the interval (s_N-varepsilon ,s_N+varepsilon ) infinitely many times and through its complement at most a finite number of times. That means that s_N-varepsilonin S and hence S ot=emptyset. Clearly, s_N+varepsilon is an upper bound for S. By the LUB Axiom, let b be the least upper bound. s_N-varepsilonle ble s_N+varepsilon. By the triangle inequality, forall nge N, d(s_n,b)le d(s_n,s_N)+d(s_N,b)levarepsilon +varepsilon =2varepsilon. Therefore, s_nlongrightarrow b and so R is complete. Q.E.D.

ee also

*Dedekind cut
*Completeness (order theory)


* [http://eom.springer.de/U/u095810.htm upper and lower bounds (including the lub axiom) at Springer's Encyclopedia of Mathematics]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Least-upper-bound property — In mathematics, the least upper bound property is a fundamental property of the real numbers and certain other ordered sets. The property states that any non empty set of real numbers that has an upper bound necessarily has a least upper bound… …   Wikipedia

  • Axiom of Archimedes — The axiom of Archimedes can be stated in modern notation as follows: Let x be any real number. Then there exists a natural number n such that n > x. In field theory this statement is called the Axiom of Archimedes. The same name is also applied… …   Wikipedia

  • Axiom — This article is about logical propositions. For other uses, see Axiom (disambiguation). In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self evident or to define and… …   Wikipedia

  • Axiom of choice — This article is about the mathematical concept. For the band named after it, see Axiom of Choice (band). In mathematics, the axiom of choice, or AC, is an axiom of set theory stating that for every family of nonempty sets there exists a family of …   Wikipedia

  • Fundamental axiom of analysis — In mathematics, the fundamental axiom of analysis (or simply the fundamental axiom) states:: Every non decreasing sequence of real numbers which is bounded above tends to a limit. The fundamental axiom may be shown to be equivalent to the least… …   Wikipedia

  • Completeness axiom — In mathematics the completeness axiom, also called Dedekind completeness of the real numbers, is a fundamental property of the set R of real numbers. It is the property that distinguishes R from other ordered fields, especially from the set of… …   Wikipedia

  • 0.999... — In mathematics, the repeating decimal 0.999... (which may also be written as 0.9, , 0.(9), or as 0. followed by any number of 9s in the repeating decimal) denotes a real number that can be shown to be the number one. In other words, the symbols 0 …   Wikipedia

  • Second-order logic — In logic and mathematics second order logic is an extension of first order logic, which itself is an extension of propositional logic.[1] Second order logic is in turn extended by higher order logic and type theory. First order logic uses only… …   Wikipedia

  • 0,9 periódico — En matemáticas, 0,999... es el número decimal periódico que se demuestra denota[1] al número 1. En otras palabras, los símbolos 0,999... y 1 son dos representaciones distintas del mismo número real. Las demostraciones matemáticas de esta igualdad …   Wikipedia Español

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.