- Least upper bound axiom
The least upper bound axiom, also abbreviated as the LUB axiom, is an
axiom 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|. 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
*
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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