quizsol05 - Math 104, Solution to Quiz 5 Instructor:...

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Instructor: Guoliang Wu July 14, 2009 1. (9 points) Fill in the following table. Sequences ( - 1) n n 2 sin ( 3 ) ( - 1) n n Set of subsequential limits {∞ , -∞} { 0 , 3 / 2 , - 3 / 2 } { 0 } lim sup 3 / 2 0 lim inf -∞ - 3 / 2 0 2. (6 points) Are the following statements true or false. Please circle your answer. (a) If the sequences ( s n ) and ( t n ) are bounded, then lim sup( s n + t n ) = lim sup s n + lim sup t n . Solution: False. We only have lim sup( s n + t n ) lim sup s n + lim sup t n . The idea of finding an example of strict in- equality is to let the subsequence of ( s n ) that converges to lim sup s n ‘miss’ that of ( t n ) . For instance, s n = ( - 1) n , t n = - ( - 1) n . (b) If a sequence ( s n ) is unbounded, then lim sup | s n | = . Solution: True. Suppose not. Then lim sup | s n | = s < . Thus, for ± = 1 > 0 , there exists N 0 > 0 such that | sup {| s n | : n > N 0 } - s | < ± = 1 . By triangle inequality,
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quizsol05 - Math 104, Solution to Quiz 5 Instructor:...

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