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.