Every function $f : (a,b) \to {\mathbb R}$ which is uniformly continuous on a bounded interval is bounded.  You can prove this by bounding f(x) – f(y) by breaking the total change into a bunch of small changes (using uniform continuity) just as in one common proof of the Fundamental Theorem of Calculus (this technique also gets used in the proof of Sard’s theorem, Harnack’s inequality, and some other things).
Now replace the bounded interval $(a,b)$ by a bounded subset $S \subseteq {\mathbb R}$.  Exercise: are uniformly continuous functions still bounded?