upper bound

An upper bound for a set is a ceiling: a number that no element of the set ever rises above. If you imagine the set marked on a number line, an upper bound is any point that sits at or to the right of every mark.

Precisely, a number M is an upper bound for a set S of reals if x is less than or equal to M for every x in S. Notice that M need not belong to S, and that a set with one upper bound has infinitely many, since anything larger is also an upper bound. A set that has at least one upper bound is called bounded above.

Upper bounds need not exist: the set of all natural numbers has none, because no fixed number exceeds every natural number. When upper bounds do exist, the most interesting one is the smallest, which leads to the notion of supremum.