# Supremum and Infimum

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

**Table of Contents**

## 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$ .