Least upper bound axiom

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 such. 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 $\{ x\in \mathbb{Q}|x<\sqrt{2}\}$ . 2 is certainly an upper bound for the set. However, the least upper bound for the set is $\sqrt{2}$ , which is not a rational number.

Proof that the real number line is complete

Let $\{ s_n\}_{n\in\N}$ be a Cauchy sequence. Let S be the set of real numbers that are bigger than $s n$ for only finitely many $n\in\N$ . Let $\varepsilon\in\R ^+$ . Let $N\in\N$ be such that $\forall n,m\ge N$ $|s_n-s_m|<\varepsilon$ . 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 $s_N-\varepsilon\in S$ and hence $S\not=\emptyset$ . Clearly, $s_N+\varepsilon$ is an upper bound for S. By the LUB Axiom, let b be the least upper bound. $s_N-\varepsilon\le b\le s_n+\varepsilon$ . By the triangle inequality, $d(s_n,b)\le d(s_n,s_N)+d(s_N,b)\le\varepsilon +\varepsilon =2\varepsilon$ . Therefore, $s_n\longrightarrow b$ and so $\R$ is complete. Q.E.D.

Categories: Real analysis

Last updated: 08-24-2005 21:23:34

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Proof that the real number line is complete