**LOWER AND UPPER BOUNDS**

**Upper Bound:**Let A ⊆ S, then an element s ∈ S is called upper bound of A if and only if-

a ≤ s, ∀ a ∈ A.

**Lower Bound:**Let A ⊆ S, then an element s ∈ S is called the lower bound of A is and only if-

s ≤ a, ∀ a ∈ A.

**Least Upper Bound:**If A ⊆ S and g ∈ S, then g is called the least upper bound of A if and only if the following two conditions hold-

a) g is upper bound of the set A.

b) g ≤ s for every upper bound s of A.

The least upper bound of A is denoted by

*l.u.b.A or supA.***Greatest Lower Bound:**If A ⊆ S and l ∈ S, then l is called the greatest lower bound of A if and only if the following two conditions hold-

a) l is lower bound of the set A.

b) s ≤ l, for every lower bound s of A.

The greatest upper bound of A is denoted by

*g.l.b.a. or infA.*

**SEE ALSO:**