It always exists in \mathbb{R}

It is equal to \min A

It always exists in \mathbb{Z} if A is a non empty subset of  \mathbb{Z} which is bounded below.

It is a lower bound

It is the supremum of the set of upper bounds of A

It is the infimum of the set of lower bounds of A

It is equal to -\sup(-A) where -A=\{-x | x\in A\}

It is the minimum of the set of lower bounds of A

It is the maximum of the set of upper bounds of A