Bounded Real-Valued Function on a Set

Let $X$ be a set and let $f:X\to\mathbb{R}$ be a function into the real numbers. We say that $f$ is bounded if there is a real number $M$ with $0\le M$ such that $|f(x)|\le M$ for every $x\in X$, the absolute value being that of Absolute Value in an Ordered Field for the ordered field of real numbers; such an $M$ is called a bound for $f$.