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 — 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 Snot=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.

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