104_Fall09_MT1_sol-1

104_Fall09_MT1_sol-1 - Midterm 1 for Math 104, Section 1...

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Unformatted text preview: Midterm 1 for Math 104, Section 1 Solutions 1. Prove that every real number is a limit of irrational numbers. Let x R be a real number. If x is irrational, then take the constant sequence ( x ). Then lim n x = x , so x is a limit of irrational numbers trivially. For each q Q , q + 2 /n is irrational for each n N , since if q + 2 /n = p Q , then 2 = n ( p- q ) Q , contradicting that 2 is irrational. For all > 0, by the Archimedean property, there exists N such that N > 2 / . Then for n N , | q + 2 /n- q | = 2 /n 2 /N < . Thus lim n q + 2 /n = q . So q is a limit of irrational numbers. 2. Use the intermediate value theorem to show that 2 R . Consider the polynomial f ( x ) = x 2 . Then f ( x ) is continuous. If we let = max { 1 , / (1+ | 2 x | ) } , then for y such that | y- x | < , we have | y 2- x 2 | = | ( y- x )( y + x ) | = | y- x || y + x | = | y- x || y- x +2 x | | y- x | ( | y- x | + | 2 x | ) / (1+ | 2 x | ) (1+ | 2 x | ) = by the triangle inequality and our formula for...
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104_Fall09_MT1_sol-1 - Midterm 1 for Math 104, Section 1...

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