lim sup | limit superior

Given a sequence of real numbers a_n, the supremum limit (also called the limit superior or upper limit), written limsup and pronounced 'lim-soup, ' is the limit of A_n = sup_(k>=n) a_k as n->∞, where sup_(x element S) x denotes the supremum. Note that, by definition, A_n is nonincreasing and so either has a limit or tends to -∞. For example, suppose a_n = (-1)^n/n, then for n odd, A_n = 1/(n + 1), and for n even, A_n = 1/n. Another example is a_n = sin n, in which case A_n is a constant sequence A_n = 1. When limsup a_n = liminf a_n, the sequence converges to the real number lim a_n = limsup a_n = liminf a_n.

infimum limit | limit | supremum | upper limit