Define what is meant by a uniformly continuous function $f$ on a subset $E$ of a metric space. Show that every continuous function on a closed, bounded interval is uniformly continuous. [You may assume the Bolzano-Weierstrass theorem.]

Suppose that a function $g:[0, \infty) \rightarrow \mathbb{R}$ is continuous and tends to a finite limit at $\infty$. Is $g$ necessarily uniformly continuous on $[0, \infty) ?$ Give a proof or a counterexample as appropriate.