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Unformatted text preview: Solutions to Assignment 2 Section 3.5 #4 (i) Let " > be given. Since ( x n ) and ( y n ) are Cauchy, 9 K 1 2 N such that n;m & K 1 = ) j x n ¡ x m j < " 2 ; and 9 K 2 2 N such that n;m & K 1 = ) j y n ¡ y m j < " 2 : So if we let K = max f K 1 ;K 2 g ; then if n;m & K , j ( x n + y n ) ¡ ( x m + y m ) j ¢ j x n ¡ x m j + j y n ¡ y m j < " 2 + " 2 = "; as we wanted. (ii) Let " > be given. Since ( x n ) and ( y n ) are Cauchy, 9 K 1 2 N such that n;m & K 1 = ) j x n ¡ x m j < 1 ; and 9 K 2 2 N such that n;m & K 2 = ) j y n ¡ y m j < 1 : In particular, if n & K 1 ; j x n ¡ x K 1 j < 1 and if m & K 2 ; j y m ¡ y K 2 j < 1 : Notice that this means j x n j < j x K 1 j + 1 and j y m j < j y K 2 j + 1 ; respectively, if n & K 1 and m & K 2 . Now let M 1 = j x K 1 j +1 and M 2 = j y K 2 j +1 : Then, again using the fact that ( x n ) and ( y n ) are Cauchy, 9 K 3 2 N such that n;m & K 3 = ) j x n ¡ x m j < " 2 M 2 ; and 9 K 4 2 N such that n;m & K 4 = ) j y n ¡ y m j < " 2 M 1 : Hence, if we let...
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This note was uploaded on 02/09/2011 for the course MATH 415 taught by Professor Bertrandguillou during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
 BERTRANDGUILLOU
 Linear Algebra, Algebra

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