# 9 - LECTURE 9: PROOFS FOR REAL NUMBERS AND SETS (4.3-4.5)...

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LECTURE 9: PROOFS FOR REAL NUMBERS AND SETS ( § 4.3-4.5) The human race has one really eﬀective weapon, and that is laughter. Mark Twain (1835 - 1910) 1. Inequalities We can assume some basic facts about inequalities. Let a,b R . We know a 2 0. Also, if a 0 then a n 0 for all n N . We also know ab 0 if and only if either a,b 0 or a,b 0. Multiplying by a positive number preserves an inequality, multiplying by a negative number reverses an inequality. Proof technique: Simplify inequalities and then try to rearrange them. Often factoring is involved. Result. If r R and 0 < r < 1 then 1 r (1 - r ) 4 . Scratch: Notice that r and 1 - r are positive. So, we can multiply by sides by r (1 - r ) to get 1 4 r (1 - r ) = - 4 r 2 + 4 r . Then 0 ≥ - 4 r 2 + 4 r - 1 = - (4 r 2 - 4 r + 1) = - (2 r - 1) 2 which is always true. We are now ready to do the proof. Proof. Let r R and 0 < r < 1. Notice that r (1 - r ) > 0. We have

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## This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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9 - LECTURE 9: PROOFS FOR REAL NUMBERS AND SETS (4.3-4.5)...

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