Supremum

analysis

Definition

Supremum

The smallest upper bound $S$ of a sequence is called the supremum. Thus, the supremum $\sup a_n$ is a real number $S_0$, such that the following properties are fulfilled:

1. $\forall n\in \mathbb{N}:a_n\le S_0$
2. $\forall n\in \mathbb{N}:a\le S⟹S_0\le S$

If the sequence is not bounded above, $\sup a_n=\infty$.

A supremum is called maximum if it is an element of sequence.