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Every subset of  is a set of real numbers. We shall define upper and lower bounds for a non-empty set of real numbers.
Upper bound. If for a set  of real  such that , then  is said to be an upper bound of . In such a case is said to be bounded above. If there is a least member amongst the upper bounds of the set , then this member is called least upper bound (l.u.b) or supremum of the set , it is usually denoted by  .
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  .
It follows that if  has at least one lower bound then there are infinitely many lower bounds of  less than it. In case  has no lower bound,  is said to be unbounded below.
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.
For every bounded set  such that  . If  is unbounded then there exists no such .
Greatest and Least Members of Sets of Real. A number is said to be the greatest (or largest) member of a set  if . If such a number exists, then it is unique and is also the supremum of the set . A set may or may not have a greatest member such as has  as the greatest member, but  has no greatest member.
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.
Examples:
- The set
 is bounded below, and unbounded above.
- The set
 is an unbounded set.
- Spremum and infimum for a set, if exist, are unique.
- The null set is neither bounded below or above, nor unbounded.
If  , then  and  .
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