inf | lim inf | limit inferior

Given a sequence of real numbers a_n, the infimum limit (also called the limit inferior or lower limit), written lim inf and pronounced 'lim-inf, ' is the limit of A_n = inf_(k>n) a_k as n->∞. Note that by definition, A_n is nondecreasing, 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, and for n even, A_n = - 1/(n + 1). Another example is a_n = sin n, in which case A_n is a constant sequence A_n = - 1. When lim sup a_n = lim inf a_n, the sequence converges to the real number lim a_n = lim sup a_n = lim inf a_n.

infimum | limit | lower limit | supremum