Every subset of is a set of real numbers. We shall define upper and lower bounds for a non-empty set of real numbers.
It easily follows that if a set has at least one upper bound then there are infinitely many upper bounds greater than it. In case has no upper bound, is said to be unbounded above.
Lower bound. If, for a set of real such that, then is said to be a lower bound of. In such a case is said to be bounded below. If there is a greatest member amongst the lower bounds of the set. then this member is called greatest lower bound (g.l.b.) or infimum of the set , and is usually denoted by .
From definitions it evidently follows that supremum and infimum of sets, if exist, are unique. The existence of supremum and infimum of non-empty sets bounded above and below respectively is ensured by the completeness axiom in. It should be noted, from definition, if is the supremum of a set then for every at least one member such that Similarly, if is the infimum of then for every at least on member such that.
Bounded and Unbounded Sets of Real. If a set of reals is bounded both above and below, then it is said to be bounded. In case is either unbounded above, or below then it is said to be unbounded. For example, the set is a bounded set and the set is an unbounded set.
Similarly, a number is said to be the least (or smallest) member of a set if . If such an exists, then, it is unique and is also the infimum of the set. A set may or may not have a least member. For example, has as the least member, but has not least member. It should be noted that a set cannot have a greatest or a least member according it is unbounded above or below.