Real numbers
The real numbers form an ordered field in which every non-empty set that is bounded above has a least upper bound.
The real numbers, , are a set with elements and where , equipped with operations and , and an ordering satisfying the following axioms:
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is commutative and associative with identity , and every has an inverse under .
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is commutative and associative with identity , and every has an inverse under .
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is distributive over .
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, exactly one of the following holds: , , , and , if and then .
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, if then , and if and then .
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Given any set of reals that is non-empty and bounded above, there exists a least upper bound of in . [This is the least upper bound axiom.]
Remarks on the axioms
From axioms (1-5) one can check for example that : otherwise , so , hence , a contradiction. The rationals sit inside by identifying with , but itself does not satisfy axiom (6): the set has no least upper bound in . In (6) it is crucial that is non-empty and bounded above; dropping either assumption destroys the existence of a least upper bound.
Existence of square roots
There exists with : let , which is non-empty and bounded above, and set , where . If then for sufficiently small , so , contradicting that is an upper bound. If then for sufficiently small , so is a smaller upper bound, contradicting . Hence . The same argument shows that exists for every and every positive .
Related
Stated in
- Definition 5.2 (Real Numbers)§5.1 Construction of