Defny Lim

Let fa n g n2N be areal sequence. (Areal sequence isafunction A: N N! Qsuchthat a n:=fA (n;i) g i2N is areal.) Define lim n!1 a n =,8 i9 N8kNja k j 2 i:

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

If every c n , then we define lim n inf a n . Remark Theconcept of lower limit and upper limit first appear in the book ( Analyse Alge'brique ) written by ...