104_Fa08_hw_3_sol

104_Fa08_hw_3_sol - Homework 3 for MATH 104 Due: Thursday...

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Unformatted text preview: Homework 3 for MATH 104 Due: Thursday Sep 25 Problem 1 Pugh, p.118, #32 Solution. (a) We claim that if p n p , then q n p . Let > 0. Then there exists N such that for all n N , d ( p n , p ) < . Let M = max { f- 1 (1) , . . . , f- 1 ( N ) } . Then for m M we have f ( n ) f ( f- 1 ( k )) = k for all k N- 1, hence for m M we have f ( n ) N and thus d ( q m , p ) < . (b) A similar argument shows that q n p if f is an injection. In this case, we chose M = max f- 1 ( { 1 , . . . , N } ), where M = 1 if f- 1 ( { 1 , . . . , N } ) is empty. M exists since the injectivity of f implies that f- 1 ( { 1 , . . . , N } ) is finite. (c) If f is a surjection, q n does not necessarily converge at all. For example, let p n = 1 / n and f ( n ) = 1 if n is odd and f ( n ) = n / 2 if n is even. Then ( q n ) = (1 , 1 , 1 , 1 / 2 , 1 , 1 / 3 , . . . ), which does not converge. Problem 2 Pugh, p.48, #36 (a) Solution. We first show that the set of non-constant polynomials with integer coe ffi cients is countable. Let P n = { p : p is a polynomial of degree n } . A polynomial....
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104_Fa08_hw_3_sol - Homework 3 for MATH 104 Due: Thursday...

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