0.1 Boundedness (Real Numbers)
A set \(A \subset \mathbb{R}\) of real numbers is bounded from above if there exists a real number \(M \in \mathbb{R}\), called an upper bound of \(A\) such that \(x \leq M\) for all \(x \in A\). Similarly, \(A\) is bounded from below if there exists a real number \(m \in \mathbb{R}\), called a lower bound of \(A\) such that \(x \geq m\) for all \(x \in A\). Furthermore, we say a set \(A\) is bounded if it is bounded both from above and below.
Equivalently, a set \(A\) is bounded if \(A \subset I\) for some bounded interval \(I=[m,M]\).