Supremum Limit

Given a sequence of real numbers , the supremum limit (also called the limit superior or upper limit), written  and pronounced 'lim-soup,' is the limit of

as , where  denotes the supremum. Note that, by definition,  is nonincreasing and so either has a limit or tends to . For example, suppose , then for  odd, , and for  even, . Another example is , in which case  is a constant sequence .

When , the sequence converges to the real number

Otherwise, the sequence does not converge.