# Least upper bound axiom

The

**least upper bound axiom**, also abbreviated as the**LUB axiom**, is anaxiom ofreal analysis stating that if a nonemptysubset of thereal numbers has anupper bound , then it has aleast 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 ofmathematics , it can be proven fromZermelo-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 — 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 $forall\; n,mge\; N,$ $|s\_n-s\_m|math>.\; So,\; the\; sequence\; passes\; through\; theinterval$ (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$.\; By\; thetriangle\; inequality,$ 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***

supremum

*Dedekind cut

*Completeness (order theory) **References*** [

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

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