Least upper bound axiom
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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. A perfect example is 
. 2 is certainly an upper bound for the set. However, this set has no least upper bound — for any 
, we can find a 
with y > x.
[edit] Proof that the real number line is complete
Let 
be a Cauchy sequence. Let S be the set of real numbers that are bigger than sn for only finitely many 
. Let 
. Let 
be such that 
. So, the sequence passes through the interval 
infinitely many times and through its complement at most a finite number of times. That means that 
and hence 
. Clearly, 
is an upper bound for S. By the LUB Axiom, let b be the least upper bound. 
. By the triangle inequality, 
. Therefore, 
and so 
is complete. Q.E.D.
