Infimum Limit
Given a sequence of real numbers a_n, the infimum limit (also called the limit inferior or lower limit), written liminf and pronounced 'lim-inf,' is the limit of
| A_n=inf_(k>n)a_k |
as n->infty. Note that by definition, A_n is nondecreasing, and so either has a limit or tends to infty. 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=sinn, in which case A_n is a constant sequence A_n=-1.
When limsupa_n=liminfa_n, the sequence converges to the real number
| lima_n=limsupa_n=liminfa_n. |
Otherwise, the sequence does not converge.
See also
Infimum, Limit, Lower Limit, SupremumThis entry contributed by Todd Rowland
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Rowland, Todd. "Infimum Limit." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/InfimumLimit.html