Bounded above

A bounded above set is a subset $A\subseteq\mathbb R$ for which there exists $M\in\mathbb R$ such that $x\le M$ for every $x\in A$; such an $M$ is called an upper bound of $A$.

This notion depends on the order axioms and is the basic hypothesis needed to define the supremum . A set can be bounded above without having a maximum .

Examples:

• $A=(0,1)$ is bounded above (for example, by $M=1$).
• $A=\mathbb R$ is not bounded above.