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# HW7 - Math 23b Homework 7 Spring 2009 Due Wednesday April 8...

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Math 23b Homework 7 Spring, 2009 Due Wednesday, April 8. Be sure to write clearly, using complete sentences. Do not use abbreviations like s.t., w/, w/o, b/c, c/o, etc. In all problems you must prove that your answer is correct, even if the problem does not explicitly ask you to do so. In addition, one third of the grade on each exercise will be determined by the presentation of your argument. Even if your answer is in the end correct, you will lose points if there are irrelevant, extraneous or incorrect statements in your argument. There will be no revision for this homework. Since there is restriction on how much you can write in each problem, you should consider carefully what is essential to include, before writing your final answers. 1. Let y > x . (a) In 1 line, prove that there is a positive integer n such that n ( y - x ) > 1. (Use the Archimedean property.) (b) In no more than 10 lines, prove that there is an integer m such that m > nx m - 1. (Consider the case nx > 0 and nx 0 separately. Use the Archimedean property to find one m , and then use the Well Ordering property to find the smallest
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