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 $\inf S = - 1$.