Upper and Lower Bounds

Every subset of \mathbb{R} is a set of real numbers. We shall define upper and lower bounds for a non-empty set S of real numbers.
           
Upper bound: If for a set S of real \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 least upper bound (l.u.b) or supremum of the set S, 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 real\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 greatest lower bound (g.l.b.) or infimum of the set S, and 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 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 \mathbb{R}. It should be noted, from 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 on member x \in S such that l \leqslant x < l + \varepsilon .

Bounded and Unbounded Sets of Real: 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 Real: 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 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{\mathbb{R}^ + } is bounded below, and unbounded above.
  2. The set \mathbb{R}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 S = \left\{ { -  1,\frac{1}{2}, - \frac{1}{3}, - \frac{1}{4}, \ldots } \right\}, then \sup S = \frac{1}{2} and \inf S = - 1.

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