Infimum Limit

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


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 lim supa_n=lim infa_n, the sequence converges to the real number

 lima_n=lim supa_n=lim infa_n.

Otherwise, the sequence does not converge.

See also

Infimum, Limit, Lower Limit, Supremum

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Infimum Limit." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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