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hw4sol

hw4sol - Honors Introduction to Analysis I Homework IV...

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Honors Introduction to Analysis I Homework IV Solution February 23, 2009 Problem 1 A rational function is a ratio of two polynomials with real coefficients, R ( x ) = P ( x ) /Q ( x ) , Q = 0 . Equality between rational functions and the operations of addition and multiplication are defined similarly to the case of the usual fractions (i.e. the rational numbers). Prove that the set of rational functions with these operations is a field. You may assume standard facts about fractions and polynomials. Solution. Just check all the axioms, and take into account that the 0 polynomial is the polynomial with all the coefficients 0. Problem 2 Using only the axioms of an ordered field, prove that the field of complex numbers, with the usual operations, cannot be ordered (you need to show that it is impossible to select a subset P of “positive” complex numbers consistently with the properties listed in Theorem 2.2.2). Solution. The idea of the proof is that in an ordered field, x 2 is positive for any non-zero element x . Indeed, according to the axioms of an ordered field, either x or - x is positive. Since x 2 = ( - x ) 2 , it is always a product of two positive elements, hence itself positive. Now apply this to the complex numbers x = i (imaginary unit) and x = 1. We obtain that i 2 = - 1 and 1 2 = 1 are both positive. This is a contradiction: for example, their sum is 0, but the sum of two positive elements must be positive. Hence the field of complex numbers cannot be ordered. Problem 3 Compute the sup, inf, limsup, liminf, and all the limit points of the following sequences: x n = 1 /n + ( - 1) n x n = 1 + ( - 1) n /n x n = ( - 1) n + 1 /n + 2 sin( nπ/ 2) Solution.

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hw4sol - Honors Introduction to Analysis I Homework IV...

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