Bounded Sets, Maxima and Minima

Definition 1 (Upper and lower bounds for sets). Let \(M \subset \mathbb{R}\) be any subset of real numbers. A real number \(b\) is called an upper bound of \(M\) if \(x \leq b\) for all \(x \in M\). Analogously, \(a \in \mathbb{R}\) is called a lower bound of \(M\) if \(a \leq x\) for all \(x \in M\).

Definition 2 (Bounded from below or above for sets). Let \(M \subset \mathbb{R}\). If there is an upper bound for \(M\), then one calls the set bounded from above. If there is a lower bound for \(M\), then one calls the set bounded from below. If both properties hold, we call the set simply bounded.

Definition 3 (Maximal and minimal element of a set). Let \(M \subset \mathbb{R}\). An element \(d \in M\) is called maximal if \(x\leq d\) for all \(x \in M\). An element \(c \in M\) is called minimal if \(x\geq c\) for all \(x \in M\). If these numbers exist, one write \(\max M = d\) and \(\min M = c\).