Bounded Continuous Functions on a Metric Space

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(S,d)$ be a metric space.

(i) Bounded continuous functions. $C_{b}(S)$ is the set of functions $f:S\to\mathbb{R}$ that are continuous and bounded, that is, for which there is a real number $M\ge0$ with $|f(x)|\le M$ for every $x\in S$ (a set, as a subset of the set $\mathbb{R}^{S}$ of maps $S\to\mathbb{R}$, by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i)).