Unformatted text preview: from the deﬁnition of “ convergence” prove that the sequence (3 a n2 b n ) ∞ n =1 converges. 6. Suppose that ( a n ) ∞ n =1 and ( b n ) ∞ n =1 are sequences of real numbers that converge to real numbers L and M , respectively. a. Prove that if for all n ∈ Z + a n < b n , then L ≤ M . b. Give an explicit counterexample that shows that (under the same hypotheses as above) the conclusion L ≤ M cannot be replaced by L < M . c. Conversely suppose that ( a n ) ∞ n =1 and ( b n ) ∞ n =1 are sequences of real numbers that converge to real numbers L and M , respectively. What can you say (without proof) about the relationship between a n and b n if you know only that L < M ?...
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This note was uploaded on 04/02/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.
 Spring '07
 thieme
 Calculus

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