104_Fall09_hw_2_sol-1 - Selected Homework Solutions, Math...

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Unformatted text preview: Selected Homework Solutions, Math 104, section 1, Fall 2009 Chapter 1, Problem 19 (b) Prove that every continuous function f : [0 , 1] [0 , 1] has at least one fixed point. If f (0) = 0 or f (1) = 1, then we are done. Otherwise, f (0) > 0 and f (1) < 1. Let g ( x ) = f ( x )- x . Then g (0) > 0 and g (1) < 0. By the intermediate value theorem, g ( c ) = 0 for some c [0 , 1]. But then f ( c ) = c , so f has a fixed point. Chapter 1, Problem 36 A real number is algebraic if it is a root of a nonconstant polynomial with integer coefficients. (b) Prove that the roots of polynomials whose coefficients belong to some fixed arbitrary denu- merable set S R is denumerable. Proof. By Corollary 18, we need only show that for a fixed n , the set of roots of polynomials of degree n is denumerable. First, note that for s S , s is a root of the polynomial x- s , and therefore the set is infinite since S is denumerable. Suppose p ( x ) = a + a 1 x + + a n x n is a non-constant polynomial of degree n whose coefficients lie in S . The set of these polynomials S n [ x ] may be identified with a subset of S n +1 = { ( a ,...,a n ) : a i S } in which a i 6 = 0 for some 1 i n . Since S is denumerable, S n +1 is denumerable by Corollary 16 and the argument of Corollary 20. By Theorem 13, an infinite subset of a denumerableCorollary 16 and the argument of Corollary 20....
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104_Fall09_hw_2_sol-1 - Selected Homework Solutions, Math...

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