mat25-hw4solns

mat25-hw4solns - Math 25: Advanced Calculus UC Davis,...

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Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 — Solutions to Homework Assignment #4 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. (a) No matter how large a real number x is given, there is always a natural number n larger. Proof. Suppose that there is some x such that no natural number is larger than x . Then x is an upper bound for the set of natural numbers, which contradicts the Archimedean Property. (b) Given any positive number y , no matter how large, and any positive number x , no matter how small, there is some natural number n such that nx > y . Proof. Assume there is no such n . That is nx y for all n . Note that this is equivalent to n yx - 1 for all n , which contradicts item (a). (c) Given any positive number x , no matter how small, once can always find a fraction 1 n such that 1 n < x . Proof. Assume there is no such n . Then 1 n x for all n . Note that this is equivalent to n x - 1 for all n , which contradicts item (a). 2. The completeness axiom can be used to construct numbers whose existence we only suspected before. As an illustration of this idea, prove that the number α = sup { x R : x 2 < 2 } exists in R and satisfies α 2 = 2. (Hint: show that either of the assumptions α 2 < 2 and α 2 > 2 lead to a contradiction.) Proof. We will use the following auxiliary result: if x 2 < y 2 , then x < y where x,y > 0. To see this, note that x 2 < y 2 is equivalent to x 2 - y 2 < 0 or ( x - y )( x + y ) < 0. Since x,y > 0, we have that x + y 6 = 0, so we may multiply this inequality by (
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This note was uploaded on 10/16/2011 for the course MATH 34 taught by Professor Wiley during the Winter '11 term at UC Merced.

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mat25-hw4solns - Math 25: Advanced Calculus UC Davis,...

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