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limsup-liminf

# limsup-liminf - Limit sup and limit inf Introduction In...

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Limit sup and limit 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. Definition of limit sup and limit inf Definition Given a real sequence a n n 1 , we define 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,... , so we have b n 2 and c n 0 for all n . Example  1 n n n 1 1,2, 3,4,... , so we have b n   and c n  for all n . Example n n 1 1, 2, 3,... , so we have 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 , denoted by lim n inf a n .

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That is, lim n inf a n sup n c n .
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limsup-liminf - Limit sup and limit inf Introduction In...

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