Supremum and Infimum

To define Supremum and Infimum in real analysis, we will have to define upper and lower bounds first.

Upper and Lower Bounds

A set $A \subset \mathbb{R}$ of real numbers is bounded from above if there exists a real number $a \in \mathbb{R}$ , called an upper bound of A, such that $x \le a$ for every $x \in A$ .

Similarly, A is bounded from below if there exists $b \in \mathbb{R}$ , called a lower bound of A, such that $x \ge b$ for every $x \in A$ .

The set to which $a$ and $b$ respectively belong are called the upper and lower bounds of the set A.

The supremum of a set is its least upper bound, and the infimum is its greatest upper bound.

Supremum or Least Upper Bound

If the set of all upper bounds of set $A \subset \mathbb{R}$ has a smallest number k then k is called the supremum of the set A., represented by k= Sup(A).

Infimum or Greatest Lower Bound

If the set of all lower bounds of a set $A \subset \mathbb{R}$ has a greatest number K then K is called the infimum of set A, represented by K=Inf(A).

Both supremum and infimum, if exist , are unique for a given set $A \subset \mathbb{R}$ .

Bounded Set

A set $A \subset \mathbb{R}$ is said to be bounded if it’s bounded above as well as bounded below. When the set A is bounded, there exist two real numbers $m, \, M$ such that $m \le x \le M$ for all $x \in A$ .