nphw5 - a is not an upper bound of B(as c is the least such...

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Numbers and Polynomials Homework 5 Prove that each nonempty set of real numbers that is bounded below has a greatest lower bound. Solution Let A R be nonempty and bounded below; denote by B R the set of all of its lower bounds. Note that B is nonempty: indeed, A has lower bounds, by assumption. Note also that B is bounded above: indeed, each b B satisfies b 6 a for any a A by definition of lower bound. Accordingly, LUB applies to B : let c := lub B be its least upper bound. Claim : c is the greatest lower bound of A . We must show that: (i) c is a lower bound of A ; (ii) c is the greatest such. (i) Suppose not; that is, suppose that some a A satisfies a < c . It follows that
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Unformatted text preview: a is not an upper bound of B (as c is the least such) so some b ∈ B satisfies a < b ; however, this disqualifies b from being a lower bound of A . Thus c ( ∈ B ) is indeed a lower bound of A . (ii) If b ( ∈ B ) is a lower bound of A then b 6 c (as c is an upper bound of B ) so that c is indeed greatest among the lower bounds of A . Notice that: (i) in proving c to be a lower bound of A we used the fact that c is the least upper bound of B ; (ii) in proving that c is the greatest lower bound of A we used the fact that c is an upper bound of B . 1...
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This note was uploaded on 11/12/2011 for the course MAS 3300 taught by Professor Staff during the Fall '08 term at University of Florida.

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