# 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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