Real Numbers and the Dedekind Cut
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Axiom of completeness
Equations are omitted for technical reasons - download the original pdf
Let the set of all real numbers, [Equation], be divided into two sets L, R such that every member l of L is less than every member r of R, where neither L nor R is empty. Then there exists a number [Equation] such that every number less than [Equation] belongs to L and every number greater than [Equation] belongs to R. The number [Equation] is said to divide the set [Equation]. The number [Equation] may be a member of either L or R. If it is a member of L then it is the greatest member of L; if it is an member of R then it is the least member of R.
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Contents of
Real Numbers and the Dedekind Cut
1 Numbers as solutions to equations
2 The irrationality of root 2
3 The set of all rational numbers is dense
4 The Dedekind cut
5 The set of real numbers as an ordered field
6 The ordered field axioms
7 The sets of rational and real numbers are ordered fields.
8 Axiom of completeness
9 Bounded sets of numbers
10 Maximum and minimum
11 Bounded set
12 Least upper bound
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