Real Numbers and the Dedekind Cut
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Bounded set
Equations are omitted for technical reasons - download the original pdf
Let S be a set of real numbers. If there is a real number K such that, for every member [Equation goes here - download the original to see it.], [Equation] then S is said to be bounded above, and K is called an upper bound of S. Likewise, if there exists a k such that, for every member [Equation], [Equation], then k is called a lower bound of S. Example. In the set [Equation] [Equation] is a lower bound of S, and 1 is an upper bound of S. The set S is bounded above, but it does not have a greatest element in the set itself.
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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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