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mat25-hw8solns

# mat25-hw8solns - Math 25 Advanced Calculus UC Davis Spring...

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Math 25: Advanced Calculus UC Davis, Spring 2011 Math 25 – Solutions to Homework Assignment #8 1. (a) lim sup n →∞ a n = 1; lim inf n →∞ a n = - 1; the set of subsequential limits is {- 1 , 1 } . (b) lim sup n →∞ a n = 1; lim inf n →∞ a n = - 1; the set of subsequential limits is {- 1 , - 2 - 1 2 , 0 , 2 - 1 2 , 1 } . (c) lim sup n →∞ a n = 1; lim inf n →∞ a n = - 1; the set of subsequential limits is [0 , 1]. (d) lim sup n →∞ a n = lim n →∞ a n = lim inf n →∞ a n = e ; the set of subsequential limits is { e } . (e) lim sup n →∞ a n = lim n →∞ a n = lim inf n →∞ a n = 1; the set of subsequential limits is { 1 } . (f) lim sup n →∞ a n = lim n →∞ a n = lim inf n →∞ a n = ; the set of subsequential limits is . 2. (a) For any m N , since { a m +1 , a m +2 , . . . } ⊆ { a m , a m +1 , a m +2 , . . . } , b m +1 = sup { a m +1 , a m +2 , . . . } ≤ sup { a m , a m +1 , a m +2 , . . . } = b m . Thus, ( b m ) m =1 is nonincreasing. (b) Since (from above) ( b m ) m =1 is nonincreasing, either there exists some k such that b n = α for some α and all n k , or not. If this is the case, then lim n →∞ b n = L = α , so that b m

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