solutions2 - Solutions to Assignment 2 Section 3.5 #4 (i)...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 2

solutions2 - Solutions to Assignment 2 Section 3.5 #4 (i)...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online