mat25-hw4 - Math 25: Advanced Calculus UC Davis, Spring...

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Unformatted text preview: Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 Homework Assignment #4 Homework due: Tuesday 4/26/11 at beginning of discussion section Reading material. Read sections 1.7, 1.9, 1.10, 2.2, 2.4 in the textbook. Problems 1. Using the Archimedean Theorem, prove each of the three statements that follow the proof of that theorem in section 1.7 of the textbook. 2. The completeness axiom can be used to construct numbers whose ex- istence we only suspected before. As an illustration of this idea, prove that the number = sup { x R : x 2 < 2 } exists in R and satisfies 2 = 2. (Hint: show that either of the as- sumptions 2 < 2 and 2 > 2 lead to a contradiction.) 3. A set E of real numbers is called ultra-dense if for every real numbers a < b , the interval ( a,b ) contains infinitely many elements of E . Prove that E is ultra-dense if and only if it is dense. 4. For real numbers x,y , show that max { x,y } = ( x + y ) / 2 + | x- y | / 2....
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mat25-hw4 - Math 25: Advanced Calculus UC Davis, Spring...

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