s12 - Math 5615H: Introduction to Analysis I. Fall 2011...

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Unformatted text preview: Math 5615H: Introduction to Analysis I. Fall 2011 Homework #12. Problems and Solutions. #1. Let a j and b j,k be real numbers defined for all j,k = 0 , 1 , 2 ,... , such that X j =0 | a j | A = const < , | b j,n | B = const < for all j,n ; and lim n b j,n = 0 for each j . Show that lim n j =0 a j b j,n = 0. Solution. We always have limsup n ( A n + B n ) = lim n sup k n ( A k + B k ) lim n sup k n A k + sup k n B k = limsup n A n + limsup n B n . For fixed natural N , X j =0 a j b j,n = A n + B n , where A n := N X j =0 a j b j,n , B n := X j = N +1 a j b j,n . Here | A n | max j N | a j | N X j =0 | b j,n | A N X j =0 | b j,n | 0 as n . Therefore, limsup n fl fl fl fl X j =0 a j b j,n fl fl fl fl limsup n ( | A n | + | B n | ) limsup n | A n | + limsup n | B n | 0 + B X j = N +1 | a j | 0 as...
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s12 - Math 5615H: Introduction to Analysis I. Fall 2011...

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