Supremum/Infimum

In this sheet, $A$ is a subset of $\mathbb R$.

Definition

Definition : Supremum

The supremum is, if it exists, the lowest upper bound. It is noted as

$$\operatorname{sup}(A)$$

Definition : Infimum

The infimum is, if it exists, the higher lower bound. It is noted as

$$\operatorname{inf}(A)$$

Results

Proposition : Property of the Supremum

If $A$ is not empty and is upper bounded then it admits a supremum.

Proposition : Characterization of the Supremum

Let $a$ be a real number. The supremum of $A$ is $a$ if, and only if,

$$\forall x \in A, x \leq a \qquad \text{and} \qquad \forall \epsilon > 0, \exists x \in A, a - \epsilon < x.$$

The same ideas exist for the infimum.