Real Numbers and the Dedekind Cut
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Least upper bound
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Any number that is greater than every element of a set S may serve as an upper bound for the set; but there may be just one number among these upper bounds that is the least upper bound. The least upper bound for a set S is a number K that is an upper bound for S such that, if is any small positive number, then [Equation goes here - download the original to see it.] is greater than some member of the set S. Likewise, the greatest lower bound of S may be defined. The least upper bound is also called the supremum, denoted sup, and the greatest lower bound is also called the infimum, denoted inf. The following theorem proves that if a set is bounded then it must have a least upper bound. Theorem If S is a non-empty set of numbers that is bounded above, then S has a supremum. Proof Let S is a non-empty set of numbers that is bounded above Let [Equation] Define the sets L and R as follows: Let [Equation] if there exists an element [Equation] such that [Equation]. Let [Equation] if, for all [Equation] we have [Equation] Since either [Equation] or [Equation], then for all [Equation], either [Equation] or [Equation]. (L and R partition S into two, with the “boundary” being on the upper boundary of S.) L is not empty, for suppose [Equation] then [Equation]. R is not empty, since S is bounded above, then there exists a least one upper bound for S, and this is a member of R. All elements of L are less than any element of R, for suppose [Equation] then there exists an [Equation] such that [Equation]; but [Equation] and if [Equation]then [Equation]; hence [Equation]. By the axiom of completeness, there exists an [Equation] such that for every positive number [Equation], [Equation]. Suppose [Equation], then there exists an [Equation] such that [Equation]. Let [Equation], then [Equation] and [Equation], hence [Equation]; therefore [Equation]. Therefore, [Equation] is the least upper bound (supremum) for S . Example. The number [Equation] is the least upper bound of the set S containing all numbers x such that [Equation]. It is not contained in the set S so it is not the maximum of S. The set S does not have a maximum but it does have a supremum (least upper bound).
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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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