Let the greatest term H of a sequence be a term which is greater than all but a finite number of the terms which are equal to H. Then H is called the upper limit of the sequence. An upper limit of a series upperlim_(n->∞) S_n = (lim_(n->∞))^_ S_n = k is said to exist if, for every ϵ>0, left bracketing bar S_n - k right bracketing bar <ϵ for infinitely many values of n and if no number larger than k has this property.

limit | lower limit | supremum limit