Supremum Limit

Given a sequence of real numbers a_n, the supremum limit (also called the limit superior or upper limit), written lim sup and pronounced 'lim-soup,' is the limit of


as n->infty, where sup_(x in S)x denotes the supremum. Note that, by definition, A_n is nonincreasing 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+1), and for n even, A_n=1/n. 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, Limit, Supremum, Upper Limit

This entry contributed by Todd Rowland

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Cite this as:

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

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