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 sup a n n That is sup a n inf b n lim n n If every b n then we define lim sup a n n 2 If every c n R then sup c n n N is called the lower limit of a n denoted by lim inf a n n That is lim inf a n sup c n n n If every c n then we define lim inf a n 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 so we have b n 2 and 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 so we have b n and c n for all n which implies that lim sup a n and lim inf a n Example n n 1 1 2 3 so we have b n n and c n for all n which implies that lim sup a n and lim inf a n Relations with convergence and divergence for upper lower limit Theorem Let a n be a real sequence then a n converges if and only if the upper limit and the lower limit are real with lim sup a n lim inf a n lim a n n n n Theorem Let a n be a real sequence then we have 1 lim n sup a n a n has no upper bound 2 lim n sup a n for any M 0 there is a positive integer n 0 such that as n n 0 we have a n M 3 lim n sup a n a if and only if a given any 0 there are infinite many numbers n such that a an and b given any 0 there is a positive integer n 0 such that as n n 0 we have a n a Similarly we also have Theorem Let a n be a real sequence then we have 1 lim n inf a n a n has no lower bound 2 lim n inf a n for any M 0 there is a positive integer n 0 such that as n n 0 we have a n M 3 lim n inf a n a if and only if a given any 0 there are infinite many numbers n such that a an and b given any 0 there is a positive integer n 0 such that as n n 0 we have a n a From Theorem 2 an Theorem 3 the sequence is divergent we give the following definitios Definition Let a n be a real sequence then we have 1 If lim n sup a n then we call the sequence a n diverges to denoted by a lim n n 2 If lim n inf a n then we call the sequence a n diverges to denoted by a lim n n Theorem Let a n be a real sequence If a is a limit point of a n then we have lim inf a n a lim sup a n n n Some useful results Theorem Let a n be a real sequence then 1 lim n inf a n lim n sup a n 2 lim n inf a n lim n sup a n and lim n sup a n lim n inf a n 3 If every a n 0 and 0 lim n inf a n lim n sup a n then we have 1 1 lim sup a1n and lim inf a1n n n lim n inf a n lim n sup a n Theorem Let a n and b n be two real sequences 1 If there is a positive integer n 0 such that a n b n then we have inf a n lim inf b n and lim sup a n lim sup b n lim n n n n 2 Suppose that lim n inf a n lim n inf b n lim n sup a n lim n sup b n then lim inf a n lim inf b n n n lim inf a n b n n lim inf a n lim sup b n or lim sup a n lim inf b n n n n n sup a n b n lim n sup a n lim sup b n lim n n In particular if a n converges we have lim sup a n b n lim a lim sup b n n n n n and lim inf a n b n lim a lim inf b n n n n n 3 Suppose that lim n inf a n lim n inf b n lim n sup a n lim n sup b n and a n 0 b n 0 n then lim inf a n lim inf b n n n lim inf a n b n n lim inf a n n lim sup b n or lim inf b n n n lim sup a n n sup a n b n lim n lim sup a n n lim sup b n n In particular if a n converges we have sup a n …