Upper and Lower Bounds

Every subset of \mathbb{R} is a set of real numbers. We shall define the upper and lower bounds for a non-empty set S of real numbers.

Upper bound: If for a set S of reals \exists {\text{ }}K \in \mathbb{R} such that \forall x \in S \Rightarrow x \leqslant K, then K is said to be an upper bound of S. In such a case, S is said to be bounded above. If there is a least member amongst the upper bounds of the set S, then this member is called the least upper bound (l.u.b) or supremum of the set S, and it is usually denoted by \sup S.
It easily follows that if a set S has at least one upper bound then there are infinitely many upper bounds greater than it. In case S has no upper bound, S is said to be unbounded above.

Lower bound: If, for a set S of reals \exists {\text{ }}k \in \mathbb{R} such that \forall x \in S \Rightarrow x \geqslant k, then k is said to be a lower bound of S. In such a case, S is said to be bounded below. If there is a greatest member amongst the lower bounds of the set S, then this member is called the greatest lower bound (g.l.b.) or infimum of the set S, and it is usually denoted by \inf S.
It follows that if S has at least one lower bound then there are infinitely many lower bounds of S less than it. In case S has no lower bound, S is said to be unbounded below.

From the definitions it evidently follows that supremum and infimum of sets, if they 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 \mathbb{R}. It should be noted, from the definition, if u is the supremum of a set S then for every \varepsilon > 0{\text{ }}\exists at least one member y \in S such that u \geqslant y > u - \varepsilon . Similarly, if l is the infimum of S then for every \varepsilon > 0{\text{ }}\exists at least one member x \in S such that l \leqslant x < l + \varepsilon .

Bounded and Unbounded Sets of Reals: If a set S of reals is bounded both above and below, then it is said to be bounded. In case S is either unbounded above or below, then it is said to be unbounded. For example, the set \left\{ {1,3,11,2059} \right\} is a bounded set and the set \mathbb{R} is an unbounded set.

For every bounded set S{\text{ }}\exists {\text{ }}k \in {\mathbb{R}^ + } such that \left| x \right| \leqslant k{\text{ }}\forall x \in S. If S is unbounded then there exists no such k.

Greatest and Least Members of Sets of Reals: A number b is said to be the greatest (or largest) member of a set S if b \in S \wedge x \in S \Rightarrow x \leqslant b. If such a number b exists, then it is unique and is also the supremum of the set S. A set may or may not have a greatest member such as \left\{ {x:1 < x \leqslant 2} \right\} has 2 as the greatest member, but \left\{ {x:1 \leqslant x \leqslant 2} \right\} has no greatest member.

Similarly, a number a is said to be the least (or smallest) member of a set S if a \in S \wedge x \in S \Rightarrow x \geqslant a. If such an a exists, then it is unique and is also the infimum of the set S. A set may or may not have a least member. For example, \left\{ {x:1 \leqslant x < 2} \right\} has 1 as the least member, but \left\{ {x:1 < x \leqslant 2} \right\} has no 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{\mathbb{R}^ + } is bounded below and unbounded above.
  2. The set \mathbb{R} is an unbounded set.
  3. The spremum and infimum for a set, if they exist, are unique.
  4. The null set is neither bounded below or above, nor unbounded.

If S = \left\{ { - 1,\frac{1}{2}, - \frac{1}{3}, - \frac{1}{4}, \ldots } \right\}, then \sup S = \frac{1}{2} and \inf S = - 1.