Ordered Set
Definition 1. An order $<$ on a set $S$ is a relation with the following two properties:
(1) If $x \in S$ and $y \in S$, then one and only one of the statements $$x<y, x=y, y<x$$ is true.
(2) $S$ is transitive.
We call $S$ an ordered set if an order is defined in $S$.
Bounded
Definition 2. Suppose $S$ is an ordered set, and $E \subset S$.
(1) If there exists a $\beta \in S$ such that $x \leq \beta, \forall x \in E$, we say that $E$ is bounded above, and call $\beta$ an upper bound of $E$.
(2) If there exists a $\alpha \in S$ such that $\alpha \leq x, \forall x \in E$, we say that $E$ is bounded below, and call $\alpha$ a lower bound of $E$.
If $E$ is bounded above and below, then we call $E$ bounded.
Least Upper Bound
Definition 3. Suppose $S$ is an ordered set, $E \subset S$, and $E$ is bounded above. Suppose there exists an $\beta \in S$ with the following properties:
(1) $\beta$ is an upper bound of $E$.
(2) If $\gamma < \beta$ then $\gamma$ is not an upper bound of $E$.
Then $\beta$ is called the least upper bound of $E$ or the supremum of $E$, and we write $$\beta = \text{sup } E.$$
Greatest Lower Bound
Definition 4. Suppose $S$ is an ordered set, $E \subset S$, and $E$ is bounded below. Suppose there exists an $\alpha \in S$ with the following properties:
(1) $\alpha$ is a lower bound of $E$.
(2) If $\alpha < \gamma$ then $\gamma$ is not a lower bound of $E$.
Then $\alpha$ is called the greatest lower bound of $E$ or the infimum of $E$, and we write $$\alpha = \text{inf } E.$$
Least Upper Bound Property
Definition 5. An ordered set $S$ is said to have the least upper bound property if the following is true:
If $\emptyset \neq E \subset S$, and $E$ is bounded above, then sup $E$ exists in $S$.