Tutorial Upper and Lower Bounds


Upper and Lower Bounds


   


Every subset of R is a set of real numbers. We shall define upper and lower bounds for a non-empty setS 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 setS 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 S is unbounded then there exists no suchk.

Greatest and Least Members of Sets of Real. A numberb is said to be the greatest (or largest) member of a set S if. If such a numberb exists, then it is unique and is also the supremum of the setS. A set may or may not have a greatest member such as has 2 as the greatest member, but has no greatest member.

Similarly, a numbera is said to be the least (or smallest) member of a set S if . If such ana 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 1 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:

  1. The set is bounded below, and unbounded above.
  2. The set is an unbounded set.
  3. Spremum and infimum for a set, if exist, are unique.
  4. The null set is neither bounded below or above, nor unbounded.
If , then and .



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