Chapter 1 - Calculus Review

Chapter 1 - Calculus Review - Chapter 1 Calculus Review Pg...

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Chapter 1 – Calculus Review Pg 3 Of greatest importance to us is that the set R of real numbers is complete - in more than one sense! First, recall that a subset A of IR is said to be bounded above if there is some x IR such that a x for all a A. Any such number x is called an upper bound for A. Pg 3 The Least Upper Bound Axiom (sometimes called the completeness axiom). Any nonempty set of real numbers with an upper bound has a least upper bound. Pg 4 Least Upper Bound of a set A (notation) Pg 4 – Example 1.1 sup(- , 1) = 1 and sup{2 - (1/n) : n = 1,2, . ..} = 2. Notice, please, that sup A is not necessarily an element of A. Pg 4 An immediate consequence of the least upper bound axiom is that we also have greatest lower bounds (g.l.b.) , just by turning things around. The details are left as Exercise 1. Pg 4 – Exercise 1 If A is a nonempty subset of JR that is bounded below, show that A has a greatest lower bound. That is, show that there is a number m IR satisfying: (i) m is a lower bound for A; and (ii) if x is a lower bound for A, then x < m. [Hint: Consider the set -A = {-a: a E A} and show that m = -sup( -A) works.]

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Pf: Notice that the greatest lower bound has the “opposite” definition of the least upper bound, so we can consider the set obtained by “reflecting” the set A about the number 0. So, I
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Chapter 1 - Calculus Review - Chapter 1 Calculus Review Pg...

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