Definition:Bounded Ordered Set/Real Numbers/Definition 2

Definition
Let $\R$ be the set of real numbers.

Let $T \subseteq \R$ be a subset of $\R$ such that:
 * $\exists K \in \R: \forall x \in T: \size x \le K$

where $\size x$ denotes the absolute value of $x$.

Then $T$ is bounded in $\R$.

Also see

 * Equivalence of Definitions of Bounded Subset of Real Numbers