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: