a real number such that
:for all in . A function that is not bounded is said to be unbounded.

If is real-valued and for all in , then the function is said to be bounded (from) above by . If for all in , then the function is said to be bounded (from) below by . A real-valued function is bounded if and only if it is bounded from above and below.

An important special case is a bounded sequence, where ' is taken to be the set of natural numbers.

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