The supremum limit, also called the limit superior or upper limit, of a sequence of real numbers , written and pronounced 'lim-soup,' is the limit of

as , where denotes the supremum. Note that, by definition, is nonincreasing and so has a limit in the extended real numbers, possibly or . For example, suppose , then for odd, , and for even, . Another example is , in which case is a constant sequence .

When is finite, the sequence converges to that real number

If the two values differ, the sequence does not converge even in the extended real numbers. If they agree at or , it converges only in the extended-real sense.

See also

Infimum Limit, Limit, Supremum, Upper Limit

Portions of this entry contributed by Todd Rowland

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Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Supremum Limit." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SupremumLimit.html

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