ch8 - Supplement on lim sup and lim inf Introduction In...

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Supplement on lim sup and lim inf Introduction In order to make us understand the information more on approaches of a given real sequence a n n 1 , we give two definitions, thier names are upper limit and lower limit. It is fundamental but important tools in analysis. We do NOT give them proofs. The reader can see the book, Infinite Series by Chao Wen-Min, pp 84-103. (Chinese Version) Definition of limit sup and limit inf Definition Given a real sequence a n n 1 ,wede f ine b n sup a m : m n and c n inf a m : m n . Example 1 1 n n 1 0,2,0,2,. .. ,sowehave b n 2and c n 0 for all n . Example  1 n n n 1 1,2, 3,4,. .. ,sowehave b n  and c n − for all n . Example n n 1 1, 2, 3,. .. ,sowehave b n n and c n − for all n . Proposition Given a real sequence a n n 1 , and thus define b n and c n as the same as before. 1 b n ≠− ,and c n ≠∀ n N . 2 If there is a positive integer p such that b p  , then b n  ∀ n N . If there is a positive integer q such that c q − , then c n − ∀ n N . 3 b n is decreasing and c n is increasing. By property 3, we can give definitions on the upper limit and the lower limit of a given sequence as follows. Definition Given a real sequence a n and let b n and c n as the same as before. (1) If every b n R , then inf b n : n N is called the upper limit of a n , denoted by lim n sup a n . That is, lim n sup a n inf n b n . If every b n  , then we define lim n sup a n  . (2) If every c n R , then sup c n : n N is called the lower limit of a n
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lim n inf a n . That is, lim n inf a n sup n c n . If every c n − , then we define lim n inf a n − . Remark The concept of lower limit and upper limit first appear in the book ( Analyse Alge’brique ) written by Cauchy in 1821. But until 1882, Paul du Bois-Reymond gave explanations on them, it becomes well-known. Example 1 1 n n 1 0,2,0,2,. .. ,sowehave b n 2and c n 0 for all n which implies that lim sup a n 2 and lim inf a n 0. Example  1 n n n 1 1,2, 3,4,. .. ,sowehave b n  and c n − for all n which implies that lim sup a n  and lim inf a n − . Example
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This note was uploaded on 12/16/2011 for the course STAT 5446 taught by Professor Frade during the Fall '09 term at FSU.

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ch8 - Supplement on lim sup and lim inf Introduction In...

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